AWG connected to IN, oscilloscope connected to OUT.

IN = Sine wave of 40 Hz at 2 mVpp.

Gain of 340 still looks pretty good on the scope :)

Now how to get rid of the DC level shift. Highpass filter near 0 Hz? ~~After some research I think there is no passive highpass that will let me block 0 Hz and (near) fully allow 1 Hz. Perhaps I should focus on level shifting instead of filtering.~~

Why doesn't this work?

Tested it on breadboard but the output DC level remains the same no matter how I turn the knob. I assume because the AD620 IC is keeping it there.

Actually, I notice later that 1uF and 1M resistor might just work as a passive highpass after all:

Yes, it works! With the filter the level drops from an average of 200 mV to just 10 mV on an amplification of several hundred.