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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)< or =t identical to 1-T/T(lambda))< or =10(-4) fits of R = R(0)t(-x(b))+R(B) to the data yielded x(b) approximately equal to 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 amplitude of the prediction was reduced by a factor close to 2. The results raise the question whether the boundary conditions used in the theory should be modified.

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