Discrete breathers are found in a nonlinear one dimensional axially loaded mechanical lattice consisting of rigid links supported laterally by linear springs. We find link centered breathers for an odd number of mechanical links and pivot centered breathers where the number of links is even. Substantial parameter regions in load-frequency parameter space are found where these breathers are linearly and nonlinearly stable. This region includes the lattice in tension, in compression, and in the unloaded state. We also find that despite the rigid nature of this mechanical system both the lateral displacement and the energy-per-link are, at least, exponentially localized in the breather core.