We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotics. We first characterize the equivalence of Gaussian measures under this model. Then consistency and asymptotic normality for the maximum likelihood estimator of the microergodic parameters are established. A simulation study is presented in order to compare the finite sample behavior of the maximum likelihood estimator with the given asymptotic distribution.