  The behavior of the adjoint Wilson line in finite-temperature, $SU(2)$,
lattice gauge theory is discussed. The expectation value of the line and the
associated excess free energy reveal the response of the finite-temperature
gauge field to the presence of an adjoint source. The value of the adjoint line
at the critical point of the deconfining phase transition is highlighted. This
is not calculable in weak or strong coupling. It receives contributions from
all scales and is nonanalytic at the critical point. We determine the general
form of the free energy. It includes a linearly divergent term that is
perturbative in the bare coupling and a finite, nonperturbative piece. We use a
simple flux tube model to estimate the value of the nonperturbative piece. This
provides the normalization needed to estimate the behavior of the line as one
moves along the critical curve into the weak coupling region.
