This paper is concerned with the application of the methods of physics to statistical models for money distribution as an example of the approach of econophysics. In a closed economic system, money is conserved. By analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. Numerical simulations suggest that at least when the number of agents and the average amount of money per agent are large, this is true. The main objective of this paper is to give a rigorous proof of this result.