When you need to measure extremely small currents, even your premium DMM is not a good choice. There are specialized instruments for this, and they have resolutions down to the attoAmpere ranges. Their principle of operation is rather different from what DMMs do – instead of measuring voltages across a shunt low level picoammeters do it by compensating current using an inverting Op amp with rather extreme properties.
I wanted to measure transistor leakage currents, so I tried to make a picoamp measurement front end myself, see here:
Electronic Projects for Fun 