We have searched for a heavy charged lepton with an associated neutrino of nearly the same mass in e+e- annihilation data taken with the Mark II detector at a center-of-mass energy of 29 GeV. In order to suppress contamination from conventional two-photon reactions, this analysis uses a novel, radiative-tagging technique. Requiring the presence of an isolated, energetic photon allows a search for lepton doublets with mass splittings smaller than that previously accessible to experiment. No evidence for such a new lepton doublet has been found, enabling limits to be placed on allowed mass combinations. Mass splittings as low as 250-400 MeV/c2 are excluded for charged-lepton masses between 500 MeV/c2 and 10 GeV/c2. © 1990 The American Physical Society.