We present an algorithm for resampling a function from its values on a non-Cartesian grid onto a Cartesian grid. This problem arises in many applications such as MRI, CT, radio astronomy and geophysics. Our algorithm, termed SParse Uniform ReSampling (SPURS), employs methods from modern sampling theory to achieve a small approximation error while maintaining low computational cost. The given non-Cartesian samples are projected onto a selected intermediate subspace, spanned by integer translations of a compactly supported kernel function. This produces a sparse system of equations describing the relation between the nonuniformly spaced samples and a vector of coefficients representing the projection of the signal onto the chosen subspace. This sparse system of equations can be solved efficiently using available sparse equation solvers. The result is then projected onto the subspace in which the sampled signal is known to reside. The second projection is implemented efficiently using a digital linear shift invariant (LSI) filter and produces uniformly spaced values of the signal on a Cartesian grid. The method can be iterated to improve the reconstruction results. We then apply SPURS to reconstruction of MRI data from nonuniformly spaced k-space samples. Simulations demonstrate that SPURS outperforms other reconstruction methods while maintaining a similar computational complexity over a range of sampling densities and trajectories as well as various input SNR levels.