  The spin-magnetic moment of the proton $\mu_p$ is a fundamental property of
this particle. So far $\mu_p$ has only been measured indirectly, analysing the
spectrum of an atomic hydrogen maser in a magnetic field. Here, we report the
direct high-precision measurement of the magnetic moment of a single proton
using the double Penning-trap technique. We drive proton-spin quantum jumps by
a magnetic radio-frequency field in a Penning trap with a homogeneous magnetic
field. The induced spin-transitions are detected in a second trap with a strong
superimposed magnetic inhomogeneity. This enables the measurement of the
spin-flip probability as a function of the drive frequency. In each measurement
the proton's cyclotron frequency is used to determine the magnetic field of the
trap. From the normalized resonance curve, we extract the particle's magnetic
moment in units of the nuclear magneton $\mu_p=2.792847350(9)\mu_N$. This
measurement outperforms previous Penning trap measurements in terms of
precision by a factor of about 760. It improves the precision of the forty year
old indirect measurement, in which significant theoretical bound state
corrections were required to obtain $\mu_p$, by a factor of 3. By application
of this method to the antiproton magnetic moment $\mu_{\bar{p}}$ the fractional
precision of the recently reported value can be improved by a factor of at
least 1000. Combined with the present result, this will provide a stringent
test of matter/antimatter symmetry with baryons.
