The simplest way to read a 4-20 mA output with the Arduino Uno is to use a 250 Ohm resistor to convert the current signal into a voltage signal, which you measure using the analogRead function. Calibration is usually required, but roughly speaking, 4 mA produces 1V and 20 mA produces 5V.
And this is a very good idea, as sooner or later you'll either power down the Arduino with the external supply on, or you'll disconnect the 250 ohm resistor(inadvertently, I hope), and the Arduino will go 'pop'. At that point, you might still have another analog input that works, or not, and you might have an Arduino that works, or not. Better to put in the 10k resistor (in the blue trace @jremington drew).