Op amps with 1000 gain and 10 volt out
There are many op-amps that can put out 10V.  The LM321 is just one example. If you go on any of the semiconductor manufacturers' websites you can search for them parametrically.

Boosting a 100mV signal to 10V only requires a gain of 100, not 1000.  Be careful of bandwidth and especially slew rate limitations of the amplifier. (Depending on the frequency of your input signal, which is not stated.)