We consider complete minimal surfaces (c.m.s.'s) in R3 and their deformations. M1 is an deformation of M0 if M1 is a graph over M0 in an tubular neighborhood of M1 and M1 is - C1 close to M0. A c.m.s. M0 is isolated if all minimal surfaces M1, which are sufficiently small deformations of M0, are congruent to M0. Many of the classical minimal surfaces in R3 are known to be isolated ; however, no example was known of a nonisolated minimal surface.