The authors prove the existence and uniqueness of a solution to the integral equation \[ x(t) = f(t) + \int_{t_0}^t k(t,s,x(s)) ds ,\quad t\in [t_0,t_0+a], \] where the equation is fuzzy in the sense that the values of \(x\) (and \(f\) and \(k\) too) are membership functions on \(R^n\). It is assumed that the functions \(f\) and \((s,t)\mapsto k(t,s,x)\) are levelwise continuous and \(k\) is assumed to be locally Lipschitz-levelwise-continuous in its third variable. The proof uses the method of successive approximations and most of the fuzzy notions that are used are explained.