We consider the discrete breathers in one-dimensional diatomic nonlinear oscillator chains. A discrete breather in the limit of zero mass ratio, i.e., the anti-continuous limit, consists of a finite number of in-phase or anti-phase excited light particles, separated by particles at rest. Existence of the discrete breathers is proved for small mass ratio by continuation from the anti-continuous limit. We prove that a discrete breather is linearly stable if it is continued from an anti-continuous solution consisting of a single excited particle or alternating anti-phase excited particles, otherwise it is linearly unstable, near the anti-continuous limit.