  Ramsey's method of separated oscillatory fields is applied to the excitation
of the cyclotron motion of short-lived ions in a Penning trap to improve the
precision of their measured mass. The theoretical description of the extracted
ion-cyclotron-resonance line shape is derived out and its correctness
demonstrated experimentally by measuring the mass of the short-lived $^{38}$Ca
nuclide with an uncertainty of $1.6\cdot 10^{-8}$ using the ISOLTRAP Penning
trap mass spectrometer at CERN. The mass value of the superallowed beta-emitter
$^{38}$Ca is an important contribution for testing the conserved-vector-current
hypothesis of the electroweak interaction. It is shown that the Ramsey method
applied to mass measurements yields a statistical uncertainty similar to that
obtained by the conventional technique ten times faster.
