S. Vugalter and T. Weidl On the discrete spectrum of a pseudo-relativistic two-body pair operator (109K, Latex) ABSTRACT. We prove Cwikel-Lieb-Rosenbljum and Lieb-Thirring type bounds on the discrete spectrum of a two-body pair operator and calculate spectral asymptotics for the eigenvalue moments and the local spectral density in the pseudo-relativistic limit.