I've got yet another cactus rotten and this was most likely causes by overwatering. I'll try to rescue it but I'll likely need a new one anyway.
Are there perhaps cacti that don't rot when overwatered?
No, there aren't - but the trick is to make sure they are not left in standing water in any outside pot or tray. When you do water, go back after 30 minutes and empty out any water left in there, and if there's a lot, go back 30 minutes after that and do it again, until there's no more water collecting. This assumes that you have the cactus planted in a gritty, freedraining compost in a pot with drainage holes which are not blocked. And only water sparingly and, in comparison to most houseplants, infrequently.
The rot is a bacterial disease of plants (fat and no fat). So be sure you get your new cactus and plant it in a vessel disinfected. Use a new pot of clay (which allows the perspiration), no plastic and no glazed ceramic. Throw away all the old soil, put little gravel in the bottom for good drainage.
If you want re-use your old pot, if it is of clay, you can. Put it under running water and remove all residual land. Then completely immerse the pot in hot water with much bleach. Let it in immersion at least 24h. Then brush it with a stiff brush, under running water, until you have removed every trace of mold. Rinse it until you no longer feel under your fingers the slimy bleach. Let it dry upside down 2 or 3 days.
Then put the ground, good soil for cacti is: 1/3 clay, 1/3 sand, 1/3 ground of the forest. Plant your new cactus and NO WATER IT for 7-10 days. Follow the Bamboo's suggests for watering it.
NEVER water it in winter, if you want flowers (when your cactus type usually makes flowers). That means, no water from half octuber till half march. The plant appears to be dry, but do not worry, just restart the watering and in a few days it will again thriving.
At last, cacti get rotten very easy if you keep them indoor: if you can, put it on a window or so, all the year. They don't suffer low temperatures (just pay attention that they don't get cold rain or snow) =)