We calculate the thermal conductivity of electrons produced by electron-electron Coulomb scattering in a strongly degenerate electron gas taking into account the Landau damping of transverse plasmons. The Landau damping strongly reduces this conductivity in the domain of ultrarelativistic electrons at temperatures below the electron plasma temperature. In the inner crust of a neutron star at temperatures T <~ 10⁷ K this thermal conductivity completely dominates over the electron conductivity due to electron-ion (electron-phonon) scattering and becomes competitive with the the electron conductivity due to scattering of electrons by impurity ions.

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