We have measured angularly resolved and absolutely calibrated spectra of the multi-MeV electrons produced by relativistic self-channeling in a high-density gas jet. Using 200 fs laser pulses with ${P}_{L}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.2\mathrm{TW}$, we have investigated the electron spectrum dependence on the plasma electron density in the range of $3\ifmmode\times\else\texttimes\fi{}{10}^{19}--4\ifmmode\times\else\texttimes\fi{}{10}^{20}{\mathrm{cm}}^{\ensuremath{-}3}$. The experimentally obtained results are closely reproduced by three-dimensional particle-in-cell simulations. A detailed analysis shows that the self-modulated laser wake field, although active, cannot explain the experimental energy spectrum. The bulk of the fast electrons are produced by direct laser acceleration at the channel betatron resonance.