XPS 420 Audio Stops Working or Works Sometimes

The integrated on-board audio of my XPS 420 stops working. When I reboot, it doesn't work. But sometimes if I go into the BIOS (version A07) and change the audio settings it will work. Then it will stop working again. Sometimes if I turn off the computer and let it sit for awhile, it will work again. Sometimes if I disable the High Definition Audio device in Device Manager (I added an ATI Radeon card which has HDMI ports), it will work. Then it will stop working at some point.

The audio was working reliably until about 2 weeks ago. I have not installed any new software, and definitely no new hardware recently. My last new software program installation was 4 months ago (September 2012)

I've tried uninstalled the integrated audio drivers, rebooting, and letting Vista detect the audio hardware and load drivers. No audio

I've tried downloading and installing the latest SigmaTel drivers. No audio.

I've tried unplugging/plugging the speakers into the green (analog) audio jack. No audio.

I've tried plugging in headphones into the front audio jack. No audio.

I'm assuming that the results from changes I make to the BIOS, audio drivers, disabling other hardware, are all random and have nothing to do with the on-board audio problem since no one change has a reliable and lasting effect.

Actually, as I'm writing this, I'm starting to think that if a capacitor in the audio circuit went bad, then this could be causing the problem.

Any ideas?

I'm thinking my best solution is to assume the integrated on-board circuit has gone bad and just buy a separate audio card like one of the basic SoundBlaster cards. I don't need anything super-duper, I don't play games on this computer. Just need basic audio. Any recommendations?