  We calculate the effective Polyakov line action corresponding to SU(2)
lattice gauge theory on a 16^3 X 4 lattice via the "relative weights" method.
We consider a variety of lattice couplings, ranging from beta=1.2 in the
strong-coupling domain, to beta=2.3 at the deconfinement transition, in order
to study how the effective action evolves with beta. Comparison of Polyakov
line correlators computed in the effective theory and the underlying gauge
theory is used to test the validity of the effective action for beta > 1.4,
while for beta=1.2, 1.4 we can compare our effective action to the one obtained
from a low-order strong-coupling expansion. Very good agreement is found at all
couplings. We find that the effective action is given by a simple expression
bilinear in the Polyakov lines. The range of the bilinear term, away from
strong coupling, grows rapidly in lattice units as beta increases.
