This paper presents an efficient method to compute the maximum likelihood (ML) estimation of the parameters of a complex 2-D sinusoidal, with the complexity order of the FFT. The method is based on an accurate barycentric formula for interpolating band-limited signals, and on the fact that the ML cost function can be viewed as a signal of this type, if the time and frequency variables are switched. The method consists in first computing the DFT of the data samples, and then locating the maximum of the cost function by means of Newton's algorithm. The fact is that the complexity of the latter step is small and independent of the data size, since it makes use of the barycentric formula for obtaining the values of the cost function and its derivatives. Thus, the total complexity order is that of the FFT. The method is validated in a numerical example.