Step 1: What You'll Need
Solder (Lead-free solder is harder to work with but less toxic. Choose wisely.)
Würth Elektronik 693 063 020 911
Valcon 412D02F-09PC003-SV SD Card Slot (OEM Part)
A Well Ventilated Area or a Fume Extractor
What appears to be a direct replacement for the SD card slot (Valcon 412D02F-09PC003-SV SD Card Slot) is available from Toby Electronics in the UK, but shipping to my place in Detroit is almost more than the cost of two new Raspberry Pi model B's. Currently Digikey stocks a suitable replacement (Würth Elektronik 693 063 020 911) that's much more economical for those of us in the USA.
Step 2: Remove the Offending SD Card Slot
Remove and discard the old, busted, SD card slot. This is easier said than done. Anyone with some electronics experience should have no problem removing this part, but if you're new to this sort of thing, go slowly.
Once the majority of the solder has been removed with the solder wick, it should be easy to pry the old socket away from the board. Start by heating the mounting pads with the soldering iron and prying the socket away from them one at a time. Only lift the socket enough to separate the feet from the pads.
Starting on one side, gently pry up the socket... heating and de-soldering the pins as you go. Once the old socket has been removed, clean up the pads with your iron to even out any little solder messes or stray solder.