Well, that is what the DreamHost Terms of Service indicates will happen:
"At the end of the contract term, the contract will automatically renew for successive thirty (30) day periods (or three (3) or six (6) or twelve (12) or twenty-four (24) month periods, depending on length of pre-payment) indefinitely until canceled in writing or via the web panel. After the first 97 days a canceled account will be canceled before the next payment is due and no more billing will occur nor will any payment be refunded, even for unused portions."
That said, DreamHost staff on these forums stated in the past, "We have no intention of holding you here against your will," and have offered a refund in at least one circumstance of which I am aware. Every circumstance is different, and DH's actions in that circumstance may not be what they are inclined to do in your situation, but it certianly doesn't hurt to ask.
If you have asked and were turned down, you might just politely request that your case be re-evaluated, possibly by a supervisor, explaining that you do not wish to continue hosting with DreamHost. It may help you get some pro-rated refund, and it might not help - but you can give it a try!