  We make a rigorous analysis of the existence and characterization of the free
boundary related to the optimal stopping problem that maximizes the mean of an
Ornstein--Uhlenbeck bridge. The result includes the Brownian bridge problem as
a limit case. The methodology hereby presented relies on a time-space
transformation that casts the original problem into a more tractable one with
an infinite horizon and a Brownian motion underneath. We comment on two
different numerical algorithms to compute the free-boundary equation and
discuss illustrative cases that shed light on the boundary's shape. In
particular, the free boundary generally does not share the monotonicity of the
Brownian bridge case.
