If you're talking about 1mV over ground being amplified to 1V over ground, and you want to use a single supply, you'll need an op-amp that includes the lower rail. Not hard to find, but you may not get this property if you don't specify it.
Perhaps if you would try ASKING rather than TELLING us to give you a circuit diagram...
As was stated above, what you need is a voltage amplifier. The total amplifier gain would need to be a gain of about 1,000, which you can either do with multiple cascaded bipolar transistor or FET stages or you can do with a couple of cascaded op amp stages.
You don't necessarily need a dual polarity rail power supply. You can do it with a single rail but you would have to capacitor couple the input and output. If you decide to go single rail and you need a 1Vrms output, you would need to have a minimum supply voltage of at least 3V.