  We show that the spectrum of a discrete two-dimensional periodic
Schr\"odinger operator on a square lattice with a sufficiently small potential
is an interval, provided the period is odd in at least one dimension. In
general, we show that the spectrum may consist of at most two intervals and
that a gap may only open at energy zero. This sharpens several results of
Kr\"uger and may be thought of as a discrete version of the Bethe--Sommerfeld
conjecture. We also describe an application to the study of two-dimensional
almost-periodic operators.
