The projection method is used to demonstrate the existence of positron attachment to three doubly excited states of helium. The e(+)He(2s(2) (1)S(e))deg, e(+)He(3s(2) (1)S(e)), and the e(+)He(2s2p (3)P(o)) states have binding energies of 0.447, 0.256, and 0.486 eV, respectively. These energies were computed with the stochastic variational method and the configuration interaction method. These states will exist as resonances in the e(+)-He continuum, and the e(+)He(2s(2) (1)S(e)) state could be detectable in the e(+)+He collision spectrum. A resonance width of 0.068 eV was computed for the e(+)He(2s(2) (1)S(e)) state by using the complex rotation method. The existence of a series of e(+)He(ns(2) (1)S(e)) resonances associated with the He(ns(2)) double Rydberg series is also predicted, and an explicit calculation demonstrating the existence of the e(+)He(3s(2) (1)S(e)) state is reported.