Rotationally quasibound levels (shape resonances) in the A 'H state of CH+ are investigated with the use of a coaxial fast-ion-beam — laser photofragment spectrometer. Individual shape-resonance levels are excited with laser radiation from high rotational levels of the X 'X+ ground state of CH+. The dissociative decay of the quasibound levels into C+( P)+H( S) is monitored by observation of the charged photofragment and measurement of its kinetic energy. Analysis of the excitation wavelength spectrum, the photofragment kinetic energies, and in some cases, the width of the shape resonances allows the assignment of the quantum numbers which characterize the quasibound levels. An improved potential-energy curve for the A 'll state is determined by the comparison of the experimental levels with those calculated for trial potentials. Thirty-two quasibound levels have been identified experimentally as grouping into A-doublet components for rotational quantum numbers 12 & J' & 35 and vibrational quantum numbers 0& v'& 10. Spectroscopic constants for the X 'X+ and A 'H states are obtained from a correlated fit of the excitation spectra for the shape-resonance levels and previous spectroscopic observations of the bound levels. Our study also allowed the first direct experimental determination of the dissociation energy of CH+: Do — 4.080+0.003 eV.