  In this paper, we prove a discrete version of the Bethe-Sommerfeld
conjecture. Namely, we show that the spectra of multi-dimensional discrete
periodic Schr\"odinger operators on $\mathbb{Z}^d$ lattice with sufficiently
small potentials contain at most two intervals. Moreover, the spectrum is a
single interval, provided one of the periods is odd, and can have a gap
whenever all periods are even.
