Let X = {X(t), t ∈ T} be a stationary centered Gaussian process with values in ℝ^d, where the parameter set T equals ℕ or ℝ₊. Let Σ_t = Cov(X₀,X_t) be the covariance function of X, and (Ω,F,P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W_t = conv{X_u, u ∈ T ∩ [0, t]} as t → +∞ and show that under the condition Σt → 0, t→∞, the rescaled convex hull (2 ln t)^-1/2W_t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ε associated to the covariance matrix Σ₀. The asymptotic behavior of the mathematical expectations Ef(W_t), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171-179, 2011].