A corner transfer matrix (CTM) is defined for the zero-field, eight-vertex model on the square lattice. Its logarithm and its diagonal form are obtained to second order in a perturbation expansion of low-temperature type. They turn out to have a very simple form, apart from certain remaindercontributions that can be ignored in the limit of a large lattice. It is conjectured that in this limit the operators have these simple forms for all temperatures less than the critical temperatureT c. The spontaneous magnetization can then easily be obtained, and agrees with the expression previously proposed. It is intended to prove some of the conjectures in subsequent papers.