After some maddening troubleshooting, including reinstalling the drivers and multiple reboots, I finally found the culprit: The Conexant SmartAudio driver doesn't work very well when it is outputting 16-bit 44.1 KHz audio; it's a lot happier outputting 24-bit 44.1 audio.
To change this in Windows 7:
Bizarre driver bug.
As an aside, choosing a higher sampling rate like 96000 Hz or 192000 Hz won't make any difference anyone besides your cat can hear, and might very well not make any audio difference at all.
Indeed, David Griesinger, creator of the legendary Lexicon reverberation sound (Griesinger's sound can be clearly heard in the "booms" at the beginning of Blade Runner's opening titles), has an amusing Powerpoint which points out that there are sound cards that advertise 96Khz sampling, but don't actually output audio above 23 Khz.
For people who don't want to download a Powerpoint file, this Google cache of the file might work.
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