Here's what's going on, and what you can (maybe) do about it:
The circuit shown assumes positive and negative supply voltages on the op-amp, with the output having positive and negative excursions. If there were no bias, you'd get exactly half the signal.
The reason you have a signal out at all is because there's about 1V of bias, and it just happens to be in the "right" direction. You can't count on this.
The reason you have bias is because Rf is too big for the setup, or because you have leakage. One giga-ohm is BIG. If you had a perfectly insulating glass cylinder with wires sticking out, and you rubbed it all over with your bare fingers, you'd have a resistor with less than one giga-ohm resistance. I suspect there's a leakage path to something, but I can't figure out what (so comments from the peanut gallery are welcome). Looking at LTC6240 packages, and the fact that you're using a ground-referenced system suggests that, if anything, leakage paths would make the signal more negative, not more positive.
The reason you see activity on the bottom trace at all is because those peaks happen when the op-amp has saturated -- in normal operation the op-amp should keep it's inverting input equal to the non-inverting input.
You have not said what you can accept for a low-frequency cutoff. You can sacrifice low-frequency performance by using a lower value of Rf (for that value of Cf, 10M-ohm would put the low-frequency cutoff at around 7Hz). You can also sacrifice overall gain by using a larger Cf -- but you'll still have the offset that you'll need to calibrate out somehow.