  We present the first measurement of the relationship between the
Sunyaev-Zel'dovich effect signal and the mass of galaxy clusters that uses
gravitational lensing to measure cluster mass, based on 14 X-ray luminous
clusters at z~0.2 from the Local Cluster Substructure Survey. We measure the
integrated Compton y-parameter, Y, and total projected mass of the clusters
(M_GL) within a projected clustercentric radius of 350 kpc, corresponding to
mean overdensities of 4000-8000 relative to the critical density. We find
self-similar scaling between M_GL and Y, with a scatter in mass at fixed Y of
32%. This scatter exceeds that predicted from numerical cluster simulations,
however, it is smaller than comparable measurements of the scatter in mass at
fixed T_X. We also find no evidence of segregation in Y between disturbed and
undisturbed clusters, as had been seen with T_X on the same physical scales. We
compare our scaling relation to the Bonamente et al. relation based on mass
measurements that assume hydrostatic equilibrium, finding no evidence for a
hydrostatic mass bias in cluster cores (M_GL = 0.98+/-0.13 M_HSE), consistent
with both predictions from numerical simulations and lensing/X-ray-based
measurements of mass-observable scaling relations at larger radii. Overall our
results suggest that the Sunyaev-Zel'dovich effect may be less sensitive than
X-ray observations to the details of cluster physics in cluster cores.
