The arbitrary control of electromagnetic waves is a key aim of photonic research. Although, for example, the control of freely propagating waves (PWs; refs 1-6) and surface waves (SWs; refs 7-10) has separately become possible using transformation optics and metamaterials, a bridge linking both propagation types has not yet been found. Such a device has particular relevance given the many schemes of controlling electromagnetic waves at surfaces and interfaces, leading to trapped rainbows(11,12), lensing(13-16), beam bending(17), deflection(18-20), and even anomalous reflection/refraction(21,22). Here, we demonstrate theoretically and experimentally that a specific gradient-index meta-surface can convert a PW to a SW with nearly 100% efficiency. Distinct from conventional devices such as prism(23) or grating(24-26) couplers, the momentum mismatch between PW and SW is compensated by the reflection-phase gradient of the meta-surface, and a nearly perfect PW-SW conversion can happen for any incidence angle larger than a critical value. Experiments in the microwave region, including both far-field and near-field characterizations, are in excellent agreement with full-wave simulations. Our findings may pave the way for many applications, including high-efficiency surface plasmon couplers, anti-reflection surfaces, light absorbers, and so on.