Change input reference to 1 - 5 V
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.