Given a set of points S = {x1 ,..., xm}⊂ ℝn and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing S. We establish that our algorithm is polynomial for fixed ε. In addition, the algorithm returns a small core set X ⊆ S, whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing X is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing S. Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate.