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Question: Do Mormons believe that Mary was still a virgin when Jesus was born?
Question: Do Mormons believe that Mary was still a virgin when Jesus was born?
Latter-day Saints believe in the virgin birth
It is claimed that Latter-day Saints believe Jesus was conceived through sexual intercourse between God the Father and Mary, and that Mary therefore was not a virgin when Jesus was born. It is also claimed that Latter-day Saints reject the "Evangelical belief" that "Christ was born of the virgin Mary, who, when the Holy Ghost came upon her, miraculously conceived the promised messiah."
Often used as evidence are a handful statements from early LDS leaders, such as Brigham Young, that directly or indirectly support this idea. However, such statements do not represent the official doctrine of the Church. The key, official doctrine of the Church is that Jesus is literally the son of God (i.e., this is not a symbolic or figurative expression), and Mary was a virgin before and after Christ's conception.
At the annunciation, Mary questioned the angel about how she could bear a child: "How shall this be, seeing I know not a man?" (Luke 1:34; the expression "know" in the Greek text is a euphemism for sexual relations). Nephi likewise described Mary as a virgin (1 Nephi 11:13-20), as did Alma1 (Alma 7:10).
Latter-day Saints believe Jesus was the Only Begotten of the Father in the flesh
[T]he testimonies of appointed witnesses leave no question as to the paternity of Jesus Christ. God was the Father of His fleshly tabernacle, and Mary, a mortal woman, was His mother. He is therefore the only person born who rightfully deserves the title “the Only Begotten Son of God.”
What the Church has not taken a position on is how the conception took place, despite speculations by various early Church leaders
The canonized scriptures are silent on how the conception took place—even Nephi's detailed vision of then-future Messiah is veiled during the part where Mary conceives (1 Nephi 11:19).
Some early leaders of the Church felt free to express their beliefs on the literal nature of God's Fatherhood of Jesus' physical body
For example, Brigham Young said the following in a discourse given 8 July 1860:
"...[T]here is no act, no principle, no power belonging to the Deity that is not purely philosophical. The birth of the Saviour was as natural as are the births of our children; it was the result of natural action. He partook of flesh and blood—was begotten of his Father, as we were of our fathers." 
But are these types of statements official Church doctrine, required for all believing Latter-day Saints to accept? No—they were never submitted to the Church for ratification or canonization. (See General authorities' statements as scripture.)
Critics have noted that this statement, and others like it, can be read to indicate there was sexual intercourse involved in the conception of Jesus. Regardless of this speculation--which goes beyond the textual data--Brigham Young's view may be seen by some contemporary Latter-day Saints as correct in that Jesus was literally physically the Son of God, just as much as any children are "of our fathers." Modern science has discovered alternative methods of conceiving children--e.g., in vitro "test tube" babies--that don't involve sexual intercourse. Thus, though processes such as artificial insemination were unknown to Brigham and thus likely not referenced by his statements, it does not necessarily follow from a modern perspective that the conception had to come about as the result of a literal sexual union. It is certainly not outside of God's power to conceive Christ by other means, while remaining his literal father. (Put another way, Jesus shared God's genetic inheritance, if you will, without necessarily requiring a sexual act to combine that inheritance with Mary's mortal contribution).
Ezra Taft Benson taught:
He was the Only Begotten Son of our Heavenly Father in the flesh—the only child whose mortal body was begotten by our Heavenly Father. His mortal mother, Mary, was called a virgin, both before and after she gave birth. (See 1 Nephi 11:20.) 
Benson's emphasis is on both the literalness of Jesus' divine birth, and the fact that Mary's virginal status persisted even immediately after conceiving and bearing Jesus.
Church leaders' statements on the literal paternity of Christ were often a reaction to various ideas which are false
- they disagreed with the tendency of conventional Christianity to deny the corporeality of God. They thus insisted that God the Father had a "natural," physical form. There was no need, in LDS theology, for a non-physical, wholly spirit God to resort to a mysterious process to conceive a Son.
- they disagreed with efforts to "allegorize" or "spiritualize" the virgin birth; they wished it understood that Christ is the literal Son of God in a physical, "natural" sense of sharing both human and divine traits in His makeup. This can be seen to be a reaction against more "liberal" strains in Christianity that saw Jesus as the literal son of Mary and Joseph, but someone endowed with God's power at some point in His life.
- they did not accept that the Father, Son, and Holy Ghost were of one "essence," but rather believed that they are distinct Personages. Thus, it is key to LDS theology that Jesus is the Son of the Father, not the Holy Ghost. To a creedal, trinitarian Christian, this might be a distinction without a difference; for an LDS Christian it is crucial.
Bruce R. McConkie said this about the birth of Christ:
God the Father is a perfected, glorified, holy Man, an immortal Personage. And Christ was born into the world as the literal Son of this Holy Being; he was born in the same personal, real, and literal sense that any mortal son is born to a mortal father. There is nothing figurative about his paternity; he was begotten, conceived and born in the normal and natural course of events, for he is the Son of God, and that designation means what it says. 
In the same volume, Elder McConkie explained his reason for his emphasis:
"Our Lord is the only mortal person ever born to a virgin, because he is the only person who ever had an immortal Father. Mary, his mother, "was carried away in the Spirit" (1 Ne. 11:13-21), was "overshadowed" by the Holy Ghost, and the conception which took place "by the power of the Holy Ghost" resulted in the bringing forth of the literal and personal Son of God the Father. (Alma 7:10; 2 Ne. 17:14; Isa. 7:14; Matt. 1:18-25; Luke 1:26-38.) Christ is not the Son of the Holy Ghost, but of the Father. (Doctrines of Salvation, vol. 1, pp. 18-20.) Modernistic teachings denying the virgin birth are utterly and completely apostate and false. 
Note that McConkie emphasized the literal nature of Christ's divinity, his direct descent from the Father, and the fact that the Holy Ghost was a tool, but not the source of Jesus' divine Parenthood.
Harold B. Lee was clear that the method of Jesus' conception had not been revealed, and discouraged speculation on the matter
Harold B. Lee said,
We are very much concerned that some of our Church teachers seem to be obsessed of the idea of teaching doctrine which cannot be substantiated and making comments beyond what the Lord has actually said.
You asked about the birth of the Savior. Never have I talked about sexual intercourse between Deity and the mother of the Savior. If teachers were wise in speaking of this matter about which the Lord has said but very little, they would rest their discussion on this subject with merely the words which are recorded on this subject in Luke 1:34-35: "Then said Mary unto the angel, How shall this be, seeing I know not a man? And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God."
Remember that the being who was brought about by [Mary's] conception was a divine personage. We need not question His method to accomplish His purposes. Perhaps we would do well to remember the words of Isaiah 55:8-9: "For my thoughts are not your thoughts, neither are your ways my ways, saith the Lord. For as the heavens are higher than the earth, so are my ways higher than your ways, and my thoughts than your thoughts."
Let the Lord rest His case with this declaration and wait until He sees fit to tell us more. 
- Ezra Taft Benson, "Five Marks of the Divinity of Jesus Christ," From a fireside address given at the University of Utah Special Events Center on 9 December 1979.
- Brigham Young, "Character of God and Christ, etc.," (8 July 1860) Journal of Discourses 8:115. (See also Brigham Young, Journal of Discourses 1:238.; Brigham Young, Journal of Discourses 4:218.; Brigham Young, Journal of Discourses 11:268..
- Ezra Taft Benson, "Joy in Christ," Ensign (March 1986), 3–4. (emphasis added) off-site
- Bruce R. McConkie, Mormon Doctrine, 2nd edition, (Salt Lake City: Bookcraft, 1966), 742. GL direct link
- Bruce R. McConkie, Mormon Doctrine, 2nd edition, (Salt Lake City: Bookcraft, 1966), 822. GL direct link
- Harold B. Lee, Teachings of Harold B. Lee (Salt Lake City, Utah: Bookcraft, 1996), 14.