Given a continuous, radial, rapidly decreasing weight v on the complex plane, we study the solid hull of its associated weighted space H_v^\infty(\mathbb C) of all the entire functions f such that v|f| is bounded. The solid hull is found for a large class of weights satisfying the condition (B) of Lusky. Precise formulations are obtained for weights of the form v(r)= exp (-ar^p), a > 0, p > 0 . Applications to spaces of multipliers are included.