The consistency and asymptotic normality of the maximum likelihood estimator is established for the first order Beta-t-GARCH model, which is a special case of the dynamic conditional score (DCS) model and closely related to the first-order Gaussian GARCH model as its limiting case. The necessary and sufficient parameter space for which the process is strictly stationary and ergodic is established, and the asymptotic results are shown to hold for the strictly stationary version of the model.