Arduino PWM controlling high-power LED

I'm wanting to use one of the Arduino's PWM output to control a high-powered LED. I can't remember the exact specifications, but it was considerably more current than the ATMega328 can provide.

I realise that this would usually be achieved with the use of a transistor, and a resistor to current-control the LED. However, I'm not wanting the LED to become less bright as the battery voltage sags, and therefore I'm wanting to use a constant current source instead of the resistor.

My first thought was to use an LM317. What I'm wanting to know, though, is if it can respond at the 64KHz it would need to?

Alternatively, could I use a simple MOSFET to achieve the same thing cheaper and simpler? (Then I wouldn't need a transistor nor constant current source, as the MOSFET would do both of these)

Thanks, Rob.