Yes. The 4xxx series, and 5xxx series, cards have a built in audio codec on the card. I had a 4870 and it came with a DVI to HDMI adapter which had t be plugged into DVI port 1 on the card. I could connect this to my TV and it passed audio. You just have to make sure that the proper audio device is selected. I believe it is ATI HD Audio codec, or something like that.
If your card did not come with a dvi to hdmi adapter, I would just read the manual to see if it needs a special adapter that will also pass audio. Obviously there are pin outs on the DVI port that provide this.
There is an amazing amount of misinformation on the web about this. About 75% of the people say Audio via DVI is impossible. One guy had the answer and said that it is certainly possible, but there is no industry standard, and that ATI etc, have developed their own standard, and thus can do it.
From the hundreds of things I've read, people are saying the special ATI DVI-HDMI adapter is necessary in order for it to work, and a straight DVI-->hdmi cable won't work, that the adapter is necessary.
But then, I'm seeing people saying that no adapter is necessary.
on newegg there are specific "DVI to HDMI adapter for 4x series cards" sold by HIS, for 9.99, but also one for 1.99 sold by Rosewill with a reviwer saying it got his audio working. I called Rosewill tech support and didn't get an answer...no wait, just did. He said.. no idea