How do we know Mary was a virgin at the time of Jesus' conception?

Matthew 1:25 obviously shows she was a virgin at least until before childbirth:

And he [St. Joseph] knew her not till she brought forth her first born son: and called his name JESUS.

St. Thomas Aquinas's commentary on Matthew explains how Matthew 1:25 does not imply she ever had carnal intercourse with her husband; thus, she remains a perpetual virgin—before, during, and after childbirth.

…lest anyone suspect that carnal intercourse occurred, it is added, And he knew her not.

In this place it should be known that this verb “to know” is taken in two senses in Sacred Scripture. Sometimes it is taken for knowledge; “Henceforth you shall know him, and you have seen him” (Jn. 14, 7). Sometimes it is taken for carnal intercourse, as in Genesis 4, 1: “And Adam knew Eve, his wife”, etc., that is, carnally.

But it is objected, why does it not say simply, he knew her not, etc., instead of, till she brought forth her son. From this it would seem to follow that he knew her afterwards. Whence, Helvidius likewise said, “Although a Virgin conceived Christ, nevertheless, afterwards she had other children of Joseph.”

And so Jerome says, that until sometimes means something limited and determinate, as if I would say, ‘I will not come until I eat, because I signify that I am about to come afterwards.’ At other times it means something unlimited and indeterminate, for example in I Corinthians 15, 25: “For he must reign, until he hath put all his enemies under his feet.” Will He not reign forever afterwards? He will indeed. But Scripture uses such manner of speaking, because it intends to remove that which could be doubted. For it could have been doubted whether He would reign when He had not put His enemies under His feet. Likewise, it could have been doubted, when the blessed Virgin had given birth, whether before the birth she had been known by Joseph. But from the start, we cannot possibly doubt; namely, because the angels sang: “Glory to God in the highest; and on earth peace to men of good will” (Lk. 2, 14). And thus, the Evangelist intends to say this. And so, Jerome argues against Helvidius: “You say, O Helvidius, that, before they came together, Joseph did not know her, because he was warned in his sleep by an angel. If, therefore, a warning in sleep was influential enough that he would not unite himself to Mary, how much more the knowledge of the angels, and the adoration of the shepherds and wise men?”* Chrysostom, however, takes knowledge for an intellectual knowing. So when it is said He knew her not, one ought to understand, namely, that he did not understand that she was of such great dignity; but after she gave birth, he knew this. Others say that it is to be taken for sensible knowledge; and their opinion is indeed sufficiently probable. For they say that Moses, from his conversing with the Lord, had so great glory in his face, that the children of Israel could not behold it (II Cor. 3, 7). Therefore, if Moses had this from his association with God, much more did this blessed Virgin, who carried Him in her womb, have so great glory in her countenance that Joseph did not know her. But the first exposition is more literal.

^{* “JEROME. Lastly, I would ask, Why then did Joseph abstain at all up to the day of birth? He will surely answer, Because of the angel’s words, That which is born in her, &c. He then who gave so much heed to a vision as not to dare to touch his wife, would he, after he had heard the shepherds, seen the Magi, and known so many miracles, dare to approach the temple of God, the seat of the Holy Ghost, the Mother of his Lord?” (Catena Aurea on St. Matthew, chap. 1 lect. 14).}

From the following verses:

Matthew 1:24-25 Then Joseph being raised from sleep did as the angel of the Lord had bidden him, and took unto him his wife: And knew her not till she had brought forth her firstborn son: and he called his name JESUS.

The phrase "knew her" is the Biblical way of saying "had sex with". Consider the following verses...

Genesis 38:26 And Judah acknowledged them, and said, She hath been more righteous than I; because that I gave her not to Shelah my son. And he knew her again no more.

JUDGES 19:25 But the men would not hearken to him: so the man took his concubine, and brought her forth unto them; and they knew her, and abused her all the night until the morning: and when the day began to spring, they let her go.

So in other words, Joseph did not have intercourse with Mary until after Jesus was born.

That proves her virginity after marriage.

Her virginity before marriage is proven by her profession to the angel Gabriel...

LUKE 1:34 Then said Mary unto the angel, How shall this be, seeing I know not a man?

...and also by Joseph's concern to take her as wife (he was afraid she was not a virgin). The angel Gabriel appeared to Joseph in a dream to dispel his fears (that she was indeed a virgin).

Matthew 1:19-20 Then Joseph her husband, being a just man, and not willing to make her a publick example, was minded to put her away privily. But while he thought on these things, behold, the angel of the Lord appeared unto him in a dream, saying, Joseph, thou son of David, fear not to take unto thee Mary thy wife: for that which is conceived in her is of the Holy Ghost.