We propose a technique for producing electron bunches that has the potential for advancing the state-of-the-art in brightness of pulsed electron sources by orders of magnitude. In addition, this method leads to femtosecond bunch lengths without the use of ultrafast lasers or magnetic compression. The electron source we propose is an ultracold plasma with electron temperatures down to 10 K, which can be fashioned from a cloud of laser-cooled atoms by photoionization just above threshold. Here we present results of simulations in a realistic setting, showing that an ultracold plasma has an enormous potential as a bright electron source.