Cosmic strings form in many extensions of the MSSM, in particular in models with an extra U(1) gauge symmetry. We propose a new production mechanism for neutralino cold dark matter in such models. We show that if the strings forming scale is less than about 10⁸ GeV, the dark matter produced from strings can overclose the universe. The mass of the new U(1) gauge boson must be much larger than the Fermi scale which makes it unobservable in upcoming accelerator experiments. In supersymmetric models with an extra U_B−L(1) gauge symmetry, the requirement of Ω ≤ 1 puts an upper bound on the neutrino mass of about 30 eV provided neutrino masses are generated by the see-saw mechanism.