  The $^2S_{1/2} - ^2P_{1/2}$ and $^2S_{1/2} - ^2P_{3/2}$ transitions in
Li-like carbon ions stored and cooled at a velocity of $\beta \approx 0.47$ in
the Experimental Storage Ring (ESR) at the GSI Helmholtz Centre in Darmstadt
have been investigated in a laser spectroscopy experiment. Resonance
wavelengths were obtained using a new continuous-wave UV laser system and a
novel extreme UV (XUV) detection system to detect forward emitted fluorescence
photons. The results obtained for the two transitions are compared to existing
experimental and theoretical data. A discrepancy found in an earlier laser
spectroscopy measurement at the ESR with results from plasma spectroscopy and
interferometry has been resolved and agreement between experiment and theory is
confirmed.
