We report measurements of the temperature difference across layers of liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures in the superfluid phase as produced by a heat current at temperatures very close to $T_{\lambda }\mathrm{}\left(X\right)\mathrm{}$. Here $X$ is the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ molar concentration that covers the range $2\times {10}^{-9\mathrm{}}<X<\mathrm{}2\mathrm{}\times {10}^{-2\mathrm{}}$. The results are analyzed in terms of contributions from the boundary and the bulk fluid resistivities $R_K$ and ${\kappa }_{\mathrm{eff}}^{-1\mathrm{}}\mathrm{}\left(X\right)\mathrm{}$. Assuming $R_K\mathrm{}\left(X\right)\mathrm{}$ to be independent of $X$, we find that for $X<\mathrm{}{10}^{-3\mathrm{}}$ there is a systematic departure from the prediction ${\kappa }_{\mathrm{eff}}\propto X^{-1\mathrm{}}$. The thermal-relaxation times are consistent with this observation. An alternative analysis invoking an anomalous boundary resistance is presented.

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