Precise knowledge of the fundamental properties of the proton is essential for our understanding of atomic structure as well as for precise tests of fundamental symmetries. We report on a direct high-precision measurement of the magnetic moment mu(p) of the proton in units of the nuclear magneton mu(N). The result, mu(p) = 2.79284734462 (+/-0.00000000082) mu(N), has a fractional precision of 0.3 parts per billion, improves the previous best measurement by a factor of 11, and is consistent with the currently accepted value. This was achieved with the use of an optimized double-Penning trap technique. Provided a similar measurement of the antiproton magnetic moment can be performed, this result will enable a test of the fundamental symmetry between matter and antimatter in the baryonic sector at the 10(-10) level.