  We present a high dispersion optical spectrum of St 34 and identify the
system as a spectroscopic binary with components of similar luminosity and
temperature (both M3+/-0.5). Based on kinematics, signatures of accretion, and
location on an H-R diagram, we conclude that St 34 is a classical T Tauri star
belonging to the Taurus-Auriga T Association. Surprisingly, however, neither
component of the binary shows LiI 6708 A, absorption, the most universally
accepted criterion for establishing stellar youth. In this uniquely known
instance, the accretion disk appears to have survived longer than the lithium
depletion timescale. We speculate that the long-lived accretion disk is a
consequence of the sub-AU separation companion tidally inhibiting, though not
preventing, circumstellar accretion. Comparisons with pre-main sequence
evolutionary models imply, for each component of St 34, a mass of 0.37+/-0.08
Msun and an isochronal age of 8+/-3 Myr, which is much younger than the
predicted lithium depletion timescale of ~ 25 Myr. Although a distance 38%
closer than that of Taurus-Auriga or a hotter temperature scale could reconcile
this discrepancy at 21-25 Myr, similar discrepancies in other systems and the
implications of an extremely old accreting Taurus-Auriga member suggest instead
a possible problem with evolutionary models. Regardless, the older age implied
by St 34's depleted lithium abundance is the first compelling evidence for a
substantial age spread in this region. Additionally, since St 34's coeval
co-members with early M spectral types would likewise fail the lithium test for
youth, current membership lists may be incomplete.
