A microscopic statistical mechanical theory of the vibrational energy relaxation of a diatomic solute in an atomic solvent is presented. The diatomic is treated as a breathing Lennard-Jones sphere. The relaxation rate is obtained from the Fourier transform of the force-force time-correlation function. The latter is expanded in powers of time (up to t(4)), and expressions for the expansion coefficients are derived using equilibrium statistical mechanics. These coefficients are used to determine the parameters of an analytic ansatz for this correlation function, which can be evaluated at all times (and thus can be Fourier transformed). The resulting theory for the time-correlation function is compared to numerical results from a molecular dynamics simulation. Theoretical results for the vibrational relaxation rate are compared to experiments on I-2 in Xe over a wide range of densities and temperatures.