Massless dark photons are predicted in hidden-sector models with an unbroken dark U(1) gauge symmetry. A particular class of these models, aiming to solve both the Yukawa-hierarchy and the dark-matter problems of the standard model, manifests natural Higgs nondecoupling properties for the dark photon. As a consequence, we show that the Higgs-boson production at colliders followed by the Higgs decay into a photon and a dark photon provides a very promising dark-photon production mechanism. This decay gives rise to an unconventional Higgs signature characterized by a resonating gamma-plus-missing-momentum system with a monochromatic photon. We discuss the sensitivity of the LHC to the corresponding signal for a Higgs boson produced in both gluon-fusion and vector-boson-fusion channels.