Using the CLEO detector at the Cornell Electron Storage Ring, we have found evidence for L=1 charmed mesons in e+e- annihilations. We observe a D**0 state at mass 2461±3±1 MeV/c2 and width 20-12-10+9+9 MeV/c2 decaying to D+-. We also see an enhancement in the m(D*+-)-m(D*+) mass-difference spectrum and by examining D*+ decay-angle dependence, we see structure consistent with two states decaying to D*+-. We observe a lower-mass state, at mass (2428±3±2) MeV/c2 and width (23-6-4+8+10) MeV/c2, assuming that the higher-mass component is from the decay D**0(2461)D*+-. We present arguments for the spin-parity assignments of these states. We also report observation of candidate L=1 cs» state decaying to D*+Ks0. © 1990 The American Physical Society.