  We consider the free boundary problem for a layer of viscous, incompressible
fluid in a uniform gravitational field, lying above a rigid bottom and below
the atmosphere. For the "semi-small" initial data, we prove the zero surface
tension limit of the problem within a local time interval. The unique local
strong solution with surface tension is constructed as the limit of a sequence
of approximate solutions to a special parabolic regularization. For the small
initial data, we prove the global-in-time zero surface tension limit of the
problem.
