The unitarity problem in curved spacetime is solved for the string des
cribed by the SL(2, R) Wess-Zumino-Witten (WZW) model. The spectrum is
computed exactly and demonstrated to be ghost-free. The new features
include (i) SL(2, R) left-right symmetry currents that have logarithmi
c cuts on the world sheet but that satisfy the usual local operator pr
oducts or commutation rules, (ii) physical states consistent with the
monodromy condition of closed strings despite the logarithmic singular
ity in the currents, and (iii) a new free boson realization for these
currents which render the SL(2, R) WZW model completely solvable.