How to read LM34 temperature sensor negative voltage with an ATmega32?
As noted in the comments, the LM34 will only be able to read temperatures in the range \$[5^\circ F, 300^\circ F]\$ configured as you have it. In this confiugration, the output voltage will never be negative so you don't have to do anything special to connect it to your ADC.

However, if you want the full temperature range, you will need to provide a negative supply rail. (See the datasheet for how to choose the resistor \$R_1\$):







simulate this circuit &ndash; Schematic created using CircuitLab

One possible solution is to simply move what you designate "ground" as, that is use a split supply for the LM34, but designate the negative rail as ground.







simulate this circuit

Note that this solution will signifcantly reduce your ADC range since now the neutral point is at \$V_s\$ rather than ground, and because the output voltage range is skewed positive (\$V_{out} \in [-0.5, 3] + V_s\$).

Another option is to add a constant voltage to the output using an op-amp circuit:







simulate this circuit

Note that \$V_{off}\$ was chosen to be 500mV since that's the lowest output voltage the LM34 can output. This type of design can be made to get the full range of your ADC by changing \$R_f\$ and \$V_{off}\$, but comes at the cost of requiring more components.

I added a unity gain inverting amplifier as the last stage since the design of the summing amplifier inverts the voltage so it's output is strictly non positive. You can omit this stage by changing what ground is defined as and doing the invert in your code.

You can also add a differential output amplifier stage instead of the summing amplifier stage, though this is usually best tied to an ADC which accepts a differential input. I'm not familiar enough with the ATMEGA32 to know if the ADC's can be configured in differential input mode or not. Since the output voltage range is skewed towards the positive rail, you're losing some of the resolution of your ADC. Differential output amplifiers usually let you set the common mode voltage level so you can set this to 1.25V to get the full ADC range, but this again adds having to compensate for the offset in code. In theory you can use two single input ADC's as a pseudo differential input ADC, though this isn't that ideal.