Inspired by the developments in photonic metamaterials, the concept of thermal metamaterials has promised new avenues for manipulating the flow of heat. In photonics, the existence of natural materials with both positive and negative permittivities has enabled the creation of metamaterials with a very wide range of effective parameters. In contrast, in conductive heat transfer, the available range of thermal conductivities in natural materials is far narrower, strongly restricting the effective parameters of thermal metamaterials and limiting possible applications in extreme environments. Here, we identify a rigorous correspondence between zero index in Maxwell's equations and infinite thermal conductivity in Fourier's law. We also propose a conductive system with an integrated convective element that creates an extreme effective thermal conductivity, and hence by correspondence a thermal analogue of photonic near-zero-index metamaterials, a class of metamaterials with crucial importance in controlling light. Synergizing the general properties of zero-index metamaterials and the specific diffusive nature of thermal conduction, we theoretically and experimentally demonstrate a thermal zero-index cloak. In contrast with conventional thermal cloaks, this meta-device can operate in a highly conductive background and the cloaked object preserves great sensitivity to external temperature changes. Our work demonstrates a thermal metamaterial which greatly enhances the capability for molding the flow of heat.