Measure low current with variable voltage
What you want is called a constant current source. You can actually build a simple one using one transistor ; but a better one uses two transistors - as in the left example below - see this page for explanations.



Here the leds would be your load, R=680 Ohm would give you about 1mA - and you would need to increase the value of the 2k2 resistor to somewhere around 22k.

The problem is that, as John Williams noted, forcing 1mA through a high resistance requires high voltage : 1MOhm would need 1kV, 10MOhm needs 10kV etc. That's very dangerous voltages! and quite some power as well : 10kV at 1mA means 10W. 

I think you need to define your needs first :


What is a tolerable error on the absolute value of current? (Would constant 1.1mA be OK?)
What is a tolerable error on the current variations? (Would current varying between 1.19mA and 1.21mA be OK?)
Do you absolutely need 1mA right from start? Or can you accept a lower current at start, growing with increased conductivity until 1mA is reached and then being held nicely constant?


If the above examples are fine, a current source as described above would do the job nicely for less than 1USD. 

If you really need a constant current source right from the start, then be prepared to work with expensive and dangerous pieces of equipment.