Analog temperature sensor with 100mV per 1 degree Fahrenheit
Is there such a thing as an analog temperature sensor with 100mv per
  degree precision (...)


I think there are sensors like this on the market, but probably sold as industrial devices (not integrated circuits) and they may be a lot more expensive than LM34 integrated circuit.


  or is there something I can stick on the end of the sensor to amplify its voltage x10?


Sure.

You can use operational amplifier in non-inverting configuration to amplify your voltage by 10. It's very simple to build.

Image from Wikipedia:



$$
V_{\text{out}} = \left(1 + \frac{ R_{\text{2}} }{ R_{\text{1}} } \right) V_{\text{in}}\!\
$$

You can use 2k\$\Omega\$ (R1) and 18k\$\Omega\$ (R2) resistors to get gain x10. Use 1% tolerance resistors if your measurement acurracy is important.

LM34 with x10 amplifier connected to output is good idea. 

Probably (it depends on voltage range you need) you can use LM358 amplifier. 

Edited:

It's low cost rail-to-rail amplifier that can work at input and output voltage = 0V, perfect for applications like this. Not all operational amplifiers can do that.