We characterize the rate-distortion function for zero-mean stationary Gaussian sources under the MSE fidelity criterion and subject to the additional constraint that the distortion is uncorrelated to the input. The solution is given by two equations coupled through a single scalar parameter. This has a structure similar to the well known water-filling solution obtained without the uncorrelated distortion restriction. Our results fully characterize the unique statistics of the optimal distortion. We also show that, for all positive distortions, the minimum achievable rate subject to the uncorrelation constraint is strictly larger than that given by the un-constrained rate-distortion function. This gap increases with the distortion and tends to infinity and zero, respectively, as the distortion tends to zero and infinity.