Circuitry to measure sub-nano ampere current
Yes there is, it's called a transimpedance amplifier. And it's fairly common to measure nA's of current. However, the limiting factor is bandwidth, amplifiers can be high bandwidth or high gain, but usually not both because if the gain is high, it band-limits the amplifier. An amplifier with high gain also amplifies the noise in the amplifier and resistors. 20dB implies that only 10x the signal is needed to measure whatever your measuring. So if you have 100pA then the top of your range would be 1nA. At some point, the resolution of the equipment that is measuring the voltage from the transimpedance amplifier is important.



These amplifiers have a high bandwidth gain product and some of which only 7fA/rtHz of input current noise (which is one of the noise sources that is going to add to the total noise budget). The noise source is in \$\frac{A}{Hz^{1/2}}\$ so if your bandwidth is 1Ghz (1nS) then that would mean the noise source would be amplified by a factor of 1000x (because 1e6Hz=1000rtHz) This would make it very difficult to make a fast measurement that you desire with this kind of bandwidth, and most likely make band limiting necessary (other noise sources not being considered which would only worsen the condition).

For voltages lower than 1uV it becomes necessary to consider all noise sources in the system and with time being considered outside of the scope of this question.  All the noise sources from the amplifier The noise floor needs to be above whatever the signal is measured, with high gain (you'd most likely need to run multi stage to get to a 0 to 5V range with the first stage having a gain of 1 million and another stage to get you to 5V.)

So in short, there is a well known method, but it would be very difficult if not impossible with the components and a DC to 1Ghz bandwidth. There is a huge difference from 1nA to 1pA in terms of noise sources. 

I did make a transimpedance amplifier that was connected to a PMT. I could see individual photons from pulses on the PMT on a 1Ghz scope, it was really cool. My bandwidth was in the MHz so I was limited on the pulses I could see.