Necessary conditions are derived for optimality of a control in a differential process with an isoperimetric constraint that is not necessarily differentiable. It is found that the gradient of the functional describing the isoperimetric constraint can be replaced by its Gateaux differential. The techniques of mathematical programming and the Dubovitskii-Milyutin theorem are employed. The necessary conditions derived are in the form of an adjoint differential equation and a pointwise maximal condition.