Do Mormons believe in the Virgin birth taught in the Bible? (LDS) According to Brigham Young, the second prophet and president of the LDS church: 


  "The birth of the Savior was as natural as 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." 
  
  (Journal of Discourses, vol. 8, p. 115).
  
  "I have given you a few leading items upon this subject, but a great
  deal more remains to be told.  Now, remember from this time forth, and
  for ever, that Jesus Christ was not begotten by the Holy Ghost."
  
  (Journal of Discourses, vol. 1, p. 51).


According to Orson Pratt the original member of the Quorum of Twelve Apostles of the Church of the Latter Day Saints: 


  The fleshly body of Jesus required a Mother as well as a Father.
  Therefore, the Father and Mother of Jesus, according to the flesh,
  must have been associated together in the capacity of Husband and
  Wife; hence the Virgin Mary must have been, for the time being, the
  lawful wife of God the Father: we use the term lawful Wife, because it
  would be blasphemous in the highest degree to say that He overshadowed
  her or begat the Savior unlawfully. 
  
  (Orson Pratt, The Seer, page 158)


According to Luke 1:26-28: 


  31 You will conceive and give birth to a son, and you are to call him Jesus. 
  
  32 He will be great and will be called the Son of the Most High. The
  Lord God will give him the throne of his father David, 
  
  33 and he will reign over Jacob’s descendants forever; his kingdom
  will never end.”
  
  34 “How will this be,” Mary asked the angel, “since I am a virgin?”
  
  35 The angel answered, “The Holy Spirit will come on you, and the
  power of the Most High will overshadow you. So the holy one to be born
  will be called[a] the Son of God. 
  
  36 Even Elizabeth your relative is
  going to have a child in her old age, and she who was said to be
  unable to conceive is in her sixth month.  37 For no word from God will
  ever fail.”


Since Brigham Young says Jesus Christ was not begotten by the Holy Ghost and Orson Pratt says Jesus required a mother and a father, does that mean the Virgin Birth spoken about in the Bible is refuted from an LDS perspective?
Yes.  Latter-Day Saints believe in the Virgin birth.

Perhaps one random sample from an official LDS Church manual called The Life and Teaching of Jesus and His Apostles Instructors Manual (Which can be found here) will help.  It says the following:


  The Significance of the Virgin Birth (A Discussion and Chalkboard
  Diagram)
  
  The teacher might wish to point out that many people in the Christian
  world want to believe in Jesus, but only as a great human being, only
  as a great man. They feel uncomfortable about the concept of the
  miraculous, virgin birth. Yet if this is denied, all of the Atonement
  must be rejected as well. It was the inheritance that came from a
  mortal mother and a divine Father that made the Atonement possible.


The confusion may come where we believe that God is his Literal, physical father.  There is no doctrine as to how this occurred.  I know of no statements of God laying with Mary.  Only that God is His Father, and Mary is His mother.  Any speculation as to how that occurred is not doctrinal, nor is it taught.  Mary was still a Virgin at Christ's birth.

Was it artificial insemination? Was it a miraculous transformation of the egg to a complete set of chromosomes? Was His zygote somehow embedded in her via the Holy Spirit?  There is no doctrine, nor even official speculation as to how this may have occurred.  Frankly I am not really even aware of any non-official speculation as to the method within the LDS faith.

When the Journal of Discourses is used as a point of LDS doctrine, one must call it into question.  These talks were not recorded except in shorthand and notes, and reconstructed by an individual for profit.  They were not produced by the LDS Church, despite the talks being by official sources.  Significant portions were never even approved of by the speakers.  

There is a whole line of research on this topic, and it is amazing what was left out and added to the talks based on the original shorthand texts.

That doesn't mean there isn't a lot of good in there, and things one can learn, but it does mean that one should probably look to LDS.org to get accurate doctrinal points rather than the JoD.

A great discussion on the JoD with modern understanding can  be found in this podcast episode.  It gives some scholarly direction as to how to look at the JoD.

Today current members may refer to the JoD, but it has been relegated to more of a curiosity and a tool than a source of doctrine.