At about 10:00 eastern time, a Grumman AA-5 lost power one minute after takeoff from Zephyrhills Municipal Airport. The pilot and one passenger were not injured; one passenger sustained minor injuries. The pilot said the preflight and engine run-up were uneventful, but the airplane failed to climb as expected after takeoff. He raised the nose and saw the tachometer indicated 1,900 to 2,000 rpm. He verified that the mixture control was full rich, the auxiliary fuel pump was on and the throttle was full forward. He advised his passengers of the situation and maneuvered the airplane for a landing in a field. Examination revealed that the mixture control cable was broken approximately three inches from the attach point at the carburetor. The mixture control arm at the carburetor was found in the idle cutoff position.