We develop a new high-performance spectral collocation method for the computation of American put and call option prices. The proposed algorithm involves a carefully posed Jacobi-Newton iteration for the optimal exercise boundary, aided by Gauss-Legendre quadrature and Chebyshev polynomial interpolation on a certain transformation of the boundary. The resulting scheme is straightforward to implement and converges at a speed several orders of magnitude faster than existing approaches. Computational effort depends on required accuracy; at precision levels similar to, say, those computed by a finite-difference grid with several hundred steps, the computational throughput of the algorithm in the Black-Scholes model is typically close to 100 000 option prices per second per CPU. For benchmarking purposes, Black-Scholes American option prices can generally be computed to ten or eleven significant digits in less than one-tenth of a second.