  In this paper, we study the Gauss map of the surfaces in the de Sitter
space-time $\mathbb S^4_1(1)$. First, we prove that a space-like surface lying
in the de Sitter space-time has pointwise 1-type Gauss map if and only if it
has parallel mean curvature vector. Then, we obtain the complete classification
of the quasi-minimal surfaces with 1-type Gauss map.
