  We analyse in detail the QED corrections to the total decay width and the
moments of the electron energy spectrum of the inclusive semi-leptonic $B \to
X_c e \nu$ decay. Our calculation includes short-distance electroweak
corrections, the complete ${\cal O}(\alpha)$ partonic terms and
leading-logarithmic QED effects up to ${\cal O}(\Lambda^3_{\rm QCD}/m_b^3)$. A
comprehensive numerical comparison of our results against those obtained with
the Monte Carlo (MC) tool PHOTOS is presented. While the comparison indicates
good overall agreement, our computation contains QED effects not included in
PHOTOS and should therefore better describe photon radiation to $B \to X_c e
\nu$ as measured by the $B$-factories. Our calculations represent the first
steps in the construction of a fully differential higher-order QED MC generator
for inclusive semi-leptonic $B$ decays.
