A noncalculus derivation of Newton's modification of Kepler's third law is given for two bodies of arbitrary masses revolving in elliptical orbits around their center of mass. Using this law and readily available astronomical data, an approximate value of the mass of the moon is calculated. At present, a precise determination of the moon's mass requires knowledge of the very small shifts of the earth from its mean orbit due to center of mass effects, although lunar probes will some day be useful for this purpose.