We present a method to estimate geocenter motion through single Low Earth Orbiter (LEO) precise orbit determination (POD) using Global Positioning System (GPS) tracking data from LEO Satellites. We fix the GPS constellation orbit ephemeris and clock biases to the “fiducial” product produced by the Jet Propulsion Laboratory (JPL). As part of the POD process of the LEO, we estimate a translation of the reference coordinate system realized by the GPS orbit and clock product. Doing so accounts for the inconsistency between the Earth’s center of mass coordinate system in which the orbit integration of the LEO is performed, and the terrestrial reference frame that the GPS orbit and clock product provides. The resulting translation parameters estimated separately from individual LEO satellites show similar variations from day to day. They represent geocenter motion, namely the difference between the origin of the terrestrial reference frame represented by the GPS orbit and clock product and the Earth’s instantaneous center of mass sensed by the LEO satellites orbital motion. Comparison with results from other techniques shows that these daily estimates of geocenter motion are robust when the LEO satellite force modeling is precise, in particular when good quality accelerometer measurements are available.