Photodiode : Transimpedance amplifier or integrator?
Please be aware that you are asking for trouble. A feedback resistance of 100MOhm will cause you all sorts of problems. You are talking such low currents that leakage will be a big problem, and if you are making a pc board, scrupulous cleanliness and removal of ALL traces of solder flux will be critical.

That said, do not use an integrator. Any problem a transimpedance amp has with leakage applies to integrators.

Furthermore, you have not described your system well enough to determine why your light levels are so low, and this opens up speculation about other potential problems. If your light levels are low because the source is far away, then you will have major problems with background and stray light. It's true you can deal with this (more optical effort, source modulation/demodulation, narrowband filters, etc), but you haven't given any indication of where to start.

I would suggest that you give serious thought to some form of optical enhancement, using a lens to focus incoming light on your detector and increase optical flux.