I solved my own problem, sort of. The trouble seems to be that there is some sort of (impedance?) problem between the TV card output and the line-in input. This causes extreme noise and distortion when the two are directly connected. (any EE types out there chime in on this?) Other devices plugged into the line-in don't do this.

There are no extra PCI slots in my SS40G for another sound card, so my Rube-Goldbergian idea was to buffer the signal through an intermediary device -- an portable MiniDisc recorder I had lying around. Hopefully I can find something a little more suited-to-task to replace the MiniDisc in the future.