We have undertaken a proper motion survey to search for pre-main-sequence stars near the Taurus-Auriga molecular cloud. Spectra were obtained for 209 possible Taurus proper motion members selected from an initial sample of 4600 stars with V less-than-or-similar-to 16 in an 8.9 square degree region. Another 68 objects selected from the less-systematic proper motion survey of Jones & Herbig [AJ, 84, 1872 (1979)] were also studied spectroscopically. Only seven new pre-main-sequence systems were identified on the basis of their strong Li I 6707 angstrom absorption. The newly discovered T Tauri stars have the same Hayashi track ages (approximately 10(6) yr) as the previously discovered pre-main-sequence population in Taurus. There is no evidence in our survey for a population of ''post-T-Tauri'' stars with ages more nearly comparable to the crossing time for the molecular cloud complex (approximately 10(7) yr). One way to reduce or eliminate this discrepancy is to assume that standard Hayashi track isochrones considerably underestimate the true ages of low-mass T Tauri stars, an idea with independent theoretical support. We suggest that the greater-than-or-similar-to 10(7) yr old stars found in the vicinity of the Taurus molecular complex by Walter et al. [Aj, 96, 297 (1988)] are low-mass members of the Cassiopeia-Taurus B star group, and constitute a distinctly older and much more widely spread stellar population than the pre-main-sequence objects seen projected against the Taurus clouds.