This is the first of two papers in which the low-temperature properties of liquid He{sup 3} are to be calculated in the thermodynamically consistent 'T-matrix' approximation. The set of coupled integral equations which are to be solved is exhibited in Part A of this paper. Part B of this paper is devoted to a preliminary, zero-temperature calculation which employs the additional approximations of using separable potentials and a noninteracting spectral function to define the interaction of two particles in the medium: the <T>{sub 0} approximation. In this approximation they obtain a spectral function for the quasi particles which they expect to display general features in common with those of the actual spectral function. Using this spectral function, they calculate the thermodynamic properties of the system and find that they compare favorably to those obtained in other calculations.