Change input reference to 1 - 5 V I am using a sensor which delivers a 4 to 20 mA output.
Converting this with a 250 Ohm resistor to ground I convert this to a 1 - 5 V output.

My question how can I change the 0V reference, so I can use the full spectrum
I found a way to change the 5V reference (analogReference()).
Is it possible to change the 0V reference of a input to 1V?

Any other solution to my problem is welcome as well! (As long as it doesn't involve I2C or anything which lowers my looptime drastically)
The ADC is ground-referenced, and there's no way to change that. The question is if that's necessary. The ADC has a 10-bit resolution, that's good for 1024 different levels between 0 and 5 V. If you convert the 4-20mA with a 250 &Omega; resistor to a 1-5 V range you still can discriminate between more than 800 different values. 

If that's not enough you can use a differential amplifier to subtract 1 V:



By choosing R3 = 1.25 R1 you convert the 0-4 V range to a 0-5 V range.