Depending on the geometry of their Fermi surfaces, Weyl semimetals and their analogs in classical systems have been classified into two types. In type-I Weyl semimetals (WSMs), the conelike spectrum at the Weyl point is not tilted, leading to a pointlike closed Fermi surface. In type-II WSMs, on the contrary, the energy spectrum around the Weyl point is strongly tilted such that the Fermi surface transforms from a point into an open surface. Here, we demonstrate, both theoretically and experimentally, a new type of (classical) Weyl semimetal whose Fermi surface is neither a point nor a surface, but a flat line. The distinctive Fermi surfaces of such semimetals, dubbed as type-III or zero-index WSMs, gives rise to unique physical properties: one of the edge modes of the semimetal exhibits a zero index of refraction along a specific direction, in stark contrast to type-I and type-II WSMs for which the index of refraction is always nonzero. We show that the zero-index response of such topological phases enables exciting applications such as extraordinary wave transmission.