Opamp would be a very simple and neat package. 1mV is below most opamps minimum voltage offset mind. The LM324 is not bad and is dirt cheap:
3mV typical offset detection.
1MHz Unity Gain Bandwidth (assuming you want say a 1,000 gain so 3mV->3V 5mv->5v etc.)...
If you use one op-amp with a gain of 1000, that would give you a total bandwidth allowance of 3KHz.
Using 2 for a gain of 900 in total (gain of 30 on each), you can get 65Khz ish max bandwidth.
3 of the op-amps in a cascade with a gain of 10 on each, you will have a total bandwidth allowance of 152Khz.
- They are cheap! Take up less space. Easier to debug instead of trying to find the faulty component.