The Whittle likelihood is a widely used and computationally efficient pseudolikelihood. However, it is known to produce biased parameter estimates with finite sample sizes for large classes of models. We propose a method for debiasing Whittle estimates for second-order stationary stochastic processes. The debiased Whittle likelihood can be computed in the same O(n log n) operations as the standard Whittle approach. We demonstrate the superior performance of our method in simulation studies and in application to a large-scale oceanographic dataset, where in both cases the debiased approach reduces bias by up to two orders of magnitude, achieving estimates that are close to those of the exact maximum likelihood, at a fraction of the computational cost. We prove that the method yields estimates that are consistent at an optimal convergence rate of n(-1/2) for Gaussian processes and for certain classes of non-Gaussian or nonlinear processes. This is established under weaker assumptions than in the standard theory, and in particular the power spectral density is not required to be continuous in frequency. We describe how the method can be readily combined with standard methods of bias reduction, such as tapering and differencing, to further reduce bias in parameter estimates.