O.K., sorry, that wasn't clear to me from your post.
The voltage drop is given by the reference voltage that the regulator uses to set its output voltage (1.25V for the LM317 - that's what you should see across the resistor).
An
LT3007, for example, has 600mV reference voltage, therefore it should be suitable for a constant current regulator with only 600mV voltage drop across the resistor defining the current.
And to anticipate the question: no, I have no personal experience with this chip.