The physical states of N = 4 conformal supergravity in four dimensions occur in twistor-string theory by Berkovits and Witten [1]. We study two alternative versions of twistor-string theory based on the B-model of weighted projective space WCP^3|2 and based on a certain construction involving open strings. The spacetime fields described by the twistor superfields contain the physical states of N = 1 conformal supergravity from above N = 4 superspace approach.