  The electron mass in atomic mass units has been determined with a relative
uncertainty of $2.8\cdot 10^{-11}$, which represents a 13-fold improvement of
the 2010 CODATA value. The underlying measurement principle combines a
high-precision measurement of the Larmor-to-cyclotron frequency ratio on a
single hydrogen-like carbon ion in a Penning trap with a corresponding very
accurate $g$-factor calculation. Here, we present the measurement results in
detail, including a comprehensive discussion of the systematic shifts and their
uncertainties. A special focus is set on the various sources of phase jitters,
which are essential for the understanding of the applied line-shape model for
the $g$-factor resonance.
