Using the covariant N=2 harmonic supergraph techniques we calculate the one-loop low-energy effective action of N=4 super-Yang–Mills theory in Coulomb branch with gauge group SU(2) spontaneously broken down to U(1). The full dependence of the low-energy effective action on both the hypermultiplet and gauge fields is determined. The direct quantum calculation confirms the correctness of the exact N=4 SYM low-energy effective action derived in <inter-ref locator="hep-th/0111062" object-type="preprint" locator-type="xxx-archive"></inter-ref> on the purely algebraic ground by invoking a hidden N=2 supersymmetry which completes the manifest N=2 one to N=4. Our results provide an exhaustive solution to the problem of finding out the exact completely N=4 supersymmetric low-energy effective action for the theory under consideration.