  We report new measurements in four cells of the thermal boundary resistance
$R$ between copper and $^4$He below but near the superfluid-transition
temperature $T_\lambda$. For $10^{-7} \leq t \equiv 1 - T/T_\lambda \leq
10^{-4}$ fits of $R = R_0 t^{x_b} + B_0$ to the data yielded $x_b \simeq 0.18$,
whereas a fit to theoretical values based on the renormalization-group theory
yielded $x_b = 0.23$. Alternatively, a good fit of the theory to the data could
be obtained if the {\it amplitude} of the prediction was reduced by a factor
close to two. The results raise the question whether the boundary conditions
used in the theory should be modified.
