The author considers the propagator between two field configurations in the quantum theory of gravity based on the new variables of Ashtekar (1986). The central tool in the analysis is the BRST formalism as developed by Batalin, Fradkin and Vilkovisky (1977, 1983). The author uses a combination of functional integral and operator techniques to show that the amplitude is expressible as the matrix element of a relatively simple ghost-extended evolution operator. The amplitude is then to be averaged over the kinematic (or gauge) group which is a semi-direct product of the spatial diffeomorphism and local SU(2) groups.