In the apostolic constitution Munificentissimus Deus, Pope Piux XII wrote that Mary, “having completed the course of her earthly life, was assumed body and soul into heavenly glory.” Thus the Assumption, a long-standing tradition of the Church, was formalized into a dogma of faith.
The wording of this quotation, however, leaves open the question you posed: Did Mary actually die before she was assumed into heaven? In fact, there is no conclusive answer. Over the centuries, arguments have been made both for and against the idea that Mary actually died first.
One argument against her death comes out of the Church’s teaching that she was conceived without original sin: if death is seen as an effect of original sin, and Mary was sinless, then many conclude she would not have had to die. On the other hand, Christ himself was free from original sin, and we know that he died, which begs the question of why Mary would be exempt from this experience if her Son were not. Many Catholics also feel uncomfortable with the notion that Mary didn’t die because they feel that it denies an important aspect of her humanity.
Bottom line: no dogmatic answer to this question has ever been given, so you are free to believe what you’d like. What’s ultimately most important is the fact that Christ honored his mother by taking her, body and soul, up to heaven – a pretty nice thing for a son to do.