Interpolation of a spatially correlated random process is used in many scientific areas. The best unbiased linear predictor, often called a kriging predictor in geostatistical science, requires the solution of a (possibly large) linear system based on the covariance matrix of the observations. In this article, we show that tapering the correct covariance matrix with an appropriate compactly supported positive definite function reduces the computational burden significantly and still leads to an asymptotically optimal mean squared error. The effect of tapering is to create a sparse approximate linear system that can then be solved using sparse matrix algorithms. Monte Carlo simulations support the theoretical results. An application to a large climatological precipitation dataset is presented as a concrete and practical illustration.