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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.