Sensing low voltages (Thermocouples)

I'm considering using my Twine to measure the exhaust gas temperatures in my wood stove, so I can hopefully be notified via text message when it needs to be stoked (or when my roommate has stoked it for me)!  In an effort to keep things simple, I am hoping to use a simple thermocouple circuit that will produce only very low voltages to be measured by my Twine via the breakout box.  The voltages produced by the thermocouple are only about 40 µV per °C, within a range of 0-800°C.  I only need a resolution of about 10°C, so hopefully I'd be able to have Twine accurately distinguish a difference of about 0.0004V at a time, between 0 and 0.03V.  Is this possible?  Or will I need to increase the complexity of this to have my voltages range between 0 and 3V?  I guess I could pull out my old EE texts and figure out how to do it, probably using resistors or something (I'm a little rusty).  

Also note that the thermocouple will produce voltage, so I don't plan on using the 3V supply in the breakout box as a part of this circuit.