Instrumentation Amplifier

There are so many out there, but here is a place to start:

http://www.analog.com/en/specialty-amplifiers/instrumentation-amplifiers/products/index.html#Instrumentation_Amplifier

Your V_out/V_in is 416.7, so get one with a max gain over that. On this list, at least 1000.

An external module for this would be SG-3016 http://www.icpdas-usa.com/sg_3016.html . The best range selection you could choose with this device is +-20 mV input with +-5V output. This would scale your 12 mV to 3V. It has adjustment resistors that can change gain about 10%. So you could not achieve 5V output. The advantage of this device that might justify the cost is that it has isolation (in case you need that).

In Mikerho's list I like the chip AD8230 because it has zero drift. I like to protect inputs (such as where you connect your resistor) with zeners to ground to prevent ESD damage.