  In a previous publication, we have constructed the Schr\"odinger functional
in Wilson's lattice QCD. It was found that the naive continuum limit leads to a
well-defined classical continuum theory. Starting from the latter, a formal
continuum definition of the Schr\"odinger functional is given and its saddle
point expansion is carried out to one-loop order of perturbation theory.
Dimensional regularization and heat kernel techniques are used to determine the
one-loop divergences. These are partly canceled by the usual renormalizations
of the quark mass and the coupling constant in QCD. An additional divergence
can be absorbed in a multiplicative renormalization of the quark boundary
fields. The corresponding boundary counterterm is a local polynomial in the
fields, so that we confirm a general expectation of Symanzik.
