Elliptical Gaussian fits are used in astronomy for accurate measurements of fundamental source parameters such as central position, peak flux density, and angular size. The full value of a noise-limited image can be realized only if the effects of noise on the fitted parameters are estimated accurately. This paper presents the equations of error propagation for two-dimensional elliptical Gaussian fits in the presence of Gaussian noise plus a new method that simplifies the use of a priori size constraints to reduce amplitude errors.