The ball-constrained weighted maximin dispersion problem P_{ball} is to find in an n-dimensional Euclidean ball the furthest point from given m points. We propose a new second-order cone programming relaxation for  P_{ball} and show that it is tight under the condition  m is less than or equal to n. We prove that  P_{ball} is NP-hard in general. Then, we propose a new approximation algorithm for solving  P_{ball}, which provides a new approximation bound of  $\frac{1-O(\sqrt{\ln(m)/n})}{2}$.