I present spectra of SN 1987K in NGC 4651 obtained 0-2 weeks past maximum brightness, and also 5-7 months past maximum. The early spectra resemble those of Type II supernovae (SNe II), with broad Hα having a well-defined P Cygni profile that appears less prominent than usual. Two weeks past maximum, the Hα emission-line and absorption-line strengths are much smaller than those in SN 1987A. At late times, no trace of the broad Hα can be found; instead, very broad emission lines of [O I] and [Ca II] dominate the spectrum. These last characteristics are typical of Type Ib supernovae long past maximum, although SNe Ib were previously thought to lack hydrogen lines at all times. The data strongly suggest, but do not prove, that SNe II and at least some SNe Ib are closely related objects. In particular, SN 1987K could have been a massive star which lost a majority of its hydrogen prior to exploding. The light curve declined more slowly after maximum than in SNe I, but not as slowly as in typical SNe II-P ("plateau"), consistent with the presence of a thin envelope. Spectra of the Type Ib SNe 1983I, 1987M, and 1988L may provide additional evidence that SNe II and SNe Ib form a continuous sequence in which the mass of the hydrogen envelope is the main variable: extremely weak Hα appears to be present in these objects near maximum light. It is also possible that SNe Ib arise from two very different types of progenitors with dissimilar explosion mechanisms. SN 1987K is proposed as the prototype of a new subclass, Type IIb, of supernovae that have the spectral characteristics of Type II near maximum brightness and those of Type Ib long past maximum.