We use the recently developed generalized double-copy procedure to construct an integrand for the fiveloop four-point amplitude of N = 8 supergravity. This construction starts from a naive double copy of the previously computed corresponding amplitude of N = 4 super-Yang-Mills theory. This is then systematically modified by adding contact terms generated in the context of the method of maximal unitarity cuts. For the simpler generalized cuts, whose corresponding contact terms tend to be the most complicated, we derive a set of formulas relating the contact contributions to the violations of the dual Jacobi identities in the relevant gauge-theory amplitudes. For more complex generalized unitarity cuts, which tend to have simpler contact terms associated with them, we use the method of maximal cuts more directly. The five-loop fourpoint integrand is a crucial ingredient towards future studies of ultraviolet properties of N = 8 supergravity at five loops and beyond. We also present a nontrivial check of the consistency of the integrand, based on modern approaches for integrating over the loop momenta in the ultraviolet region.