We completely determine the Brown measure of the sum of a self-adjoint element and an elliptic element, which is the limiting eigenvalue distribution of the random matrix \[Y_N+\sqrt{s-\frac{t}{2}}X_N+i\sqrt{\frac{t}{2}}X_N'\] where $Y_N$ is an $N\times N$ deterministic Hermitian matrix whose eigenvalue distribution converges as $N\to\infty$ and $X_N$ and $X_N'$ are independent Gaussian unitary ensembles. We also study various asymptotic behaviors of this Brown measure as the variance of the elliptic element approaches infinity.