It is shown that the elliptic algebra Aq,p(sl(2)c) has a non-trivial center at the critical level c = -2, generalizing the result of Reshetikhin and Semenov-Tian-Shansky for trigonometric algebras. A family of Poisson structures indexed by a non-negative integer k is constructed on this center. © 1997 Elsevier Science B.V.