The author proves that any spacelike immersion of \(S^{n-2}\) into the n- dimensional Minkowski space which admits some spacelike extension to the unit ball \(B\subset {\mathbb{R}}^{n-1}\) is the boundary of a maximal (with respect to the induced metric) spacelike immersion of B. In the case \(n=3\) it is shown that such a maximal surface is in general not uniquely determined by the boundary curve, however, a sufficient condition for uniqueness is given. The existence result generalizes earlier results of Bartnik-Simon on maximal graphs and uses their a priori estimates.