How do I turn off the internal speaker of my computer in Linux?

This is not the actual PC speaker, but a small, loud speaker soldered directly to mother board. All my sound comes out from both this speaker and my normal speakers connected to sound card. In GNOME's Volume Control section PCM affects both these sound sources, but section Master only the speakers connected to sound card.
For example in Windows audio comes out only from the sound card, so they are not the same device.

I'm running Fedora Core 4 if that matters.

There may be a setting for that in your bios. May have to disconnect it at the board.

The internal speaker is controlled by the pcspkr device driver.

If you don't have that loaded (or permanently compiled in to the kernel), you won't get beeps.

If it's compiled in, you'll have to recompile without it; if it's a module, you'll have to find where it's being loaded and comment it out, or put 'rmmod pcspkr' in an appropriate place.