A positron source and detector of annihilation quanta are shown to form a fast response system that senses changes in its environment. Experimental results are presented to demonstrate that such a sensor can be used to measure gas densities. A particular densitometer design has been calibrated for air at room temperature over the pressure range 100-mm Hg<p<760-mm Hg, the particular pressure range being determined by the limitations of the vacuum chamber used in the calibration. The counting rate has been observed to vary linearly with pressure. Expressions are given for the sensitivity and the statistical uncertainty as a function of source strength, counting interval, and thickness of beryllium covering the source. Experimental results are also presented which demonstrate that the presence of a solid in the field of the positrons can be detected. The observed counting rate is a function of the distance of the solid from the sensor.