Let {X-n, n >= 1} be iid elliptical random vectors in R-d, d >= 2 and let I, J be two non-empty disjoint index sets. Denote by X-n ,X- I, X-n ,X- J the subvectors of X-n with indices in I, J, respectively. For any a is an element of R-d such that a(J) is in the support of X (1, J) the conditional random sample X-n,X- I vertical bar X-n,X- J = a (J), n >= 1 consists of elliptically distributed random vectors. In this paper we investigate the relation between the asymptotic behaviour of the multivariate extremes of the conditional sample and the unconditional one. We show that the asymptotic behaviour of the multivariate extremes of both samples is the same, provided that the associated random radius of X, has distribution function in the max-domain of attraction of a univariate extreme value distribution.