The rule of thumb for shooting the moon is 1/ISO shutter speed at f/11. So if you have your camera set at ISO 400, your exposure would be 1/400 sec @ f/11. The camera's meter will be very easily fooled by the essentially pitch black sky and the bright moon surface. If you follow those guidelines, you should properly expose the moon. A similar rule of thumb is the "BSD" or Bright Sunny Day rule. If there are no clouds in the sky, you can set your exposure at 1/ISO @ f/16. This will provide a very good balanced exposure.
However if you are using only a 300mm lens, your image of the moon is going to be very small indeed in the frame. This image was shot with a D300 and 600mm Nikkor, giving it an effective focal length of 900mm. Add to that a 2x TC-300 teleconverter, and now I am shooting at an effective focal length of 1800mm. This image is hardly cropped at all, the moon fills the frame. And a very sturdy tripod is a must. I used my 14 pound Bogen studio tripod for this. Even the slightest movement and sharpness is going to take a big hit!