The growth of single-layer graphene (SLG) by chemical vapor deposition (CVD) on copper surfaces is very popular because of the self-limiting effect that, in principle, prevents the growth of few-layer graphene (FLG). However, the reproducibility of the CVD growth of homogeneous SLG remains a major challenge, especially if one wants to avoid heavy surface treatments, monocrystalline substrates and expensive equipment to control the atmosphere inside the growth system. We demonstrate here that backside tungsten coating of copper foils allows for the exclusive growth of SLG with full coverage by atmospheric pressure CVD implemented in a vacuum-free furnace. We show that the absence of FLG patches is related to the suppression of carbon diffusion through copper. In the perspective of large-scale production of graphene, this approach constitutes a significant improvement to the traditional CVD growth process since (1) a tight control of the hydrocarbon flow is no longer required to avoid FLG formation and, consequently, (2) the growth duration necessary to reach full coverage can be drastically shortened.