For a set S of n points in three dimensions, the 2-center problem asks for two congruent balls of the minimum radius, whose union covers all points of S. We first present an \(O(n^2 \log n)\) time and O(n) space algorithm for the set S of points in convex position. Our algorithm is characterized by finding a (small) set of planes such that it contains an optimal plane that partitions S into two disjoint subsets; each can be covered by a ball of the minimum radius. The convexity of S allows to efficiently enumerate all possible situations needed in computing the 2-center of S. It is further extended to solve the 2-center problem for a set of arbitrarily given points in \(O(n^2 \log n)\) time and O(n) space. This significantly improves upon two previously known results; one is a deterministic algorithm with \(O(n^{3+\epsilon })\) running time for any \(\epsilon > 0\) [Agarwal et al., SIAM J. Computing 29 (2000) 912-953], and the other is a randomized algorithm with roughly \(O(n^2 \log ^4 n \log \log n)\) expected time and \(O(n^2)\) space [Agarwal et al., Comput. Geometry 46 (2013) 734-746]. Moreover, our algorithm can be easily generalized to the 2-center problem in fixed d-dimension, \(d \ge 4\) . For small values of d, our result also gives an improvement upon the previously known \(n^{O(1)}\) time bound [Agarwal and Procopiuc, Algorithmica 33 (2002) 201-226].