Pricing Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no analytical solution is available. Pricing their American-style counterparts, which provide early exercise opportunities, poses the additional difficulty of solving a dynamic optimization problem to determine the optimal exercise strategy. We develop a numerical method for pricing American-style Asian options based on dynamic programming combined with finite-element piecewise-polynomial approximation of the value function. Numerical experiments show convergence, consistency, and efficiency. Some theoretical properties of the value function and of the optimal exercise strategy are also established.