The signal being pushed through your DVI port isn't carrying audio, all your HDMI cable is doing is taking that DVI signal through an HDMI medium and transporting it to the television.
If you want audio through your HDMI cable end to end, you'll need a video card with an HDMI port.
The ATi cards have been able to do HDMI audio via DVI since the HD2xxx series IIRC and the Sapphire
card does list it as a feature even though it also lists having two DVI ports and no HDMI port.
@OP, have you tried this