What's the point of a current source?
The current in any 2 terminal device is always the same, there's no way for a 2 terminal device to provide "extra" current.  You can't have 0A in and 4A out.

What it does do is force the current to be the given value regardless of the impedance across it.  (So therefore you can't put in an open circuit an ideal current source, the voltage would go to infinity.)

So a resistor by itself has no current through it.  Hook it across an ideal current source and you get (for your example current source) 1A through it regardless of the value of the resistor.