Exoplanets that orbit close to their host stars are much more highly irradiated than their solar system counterparts. Understanding the thermal structures and appearances of these planets requires investigating how their atmospheres respond to such extreme stellar forcing. We present spectroscopic thermal emission measurements as a function of orbital phase ("phase-curve observations") for the highly irradiated exoplanet WASP-43b spanning three full planet rotations using the Hubble Space Telescope. With these data, we construct a map of the planet's atmospheric thermal structure, from which we find large day-night temperature variations at all measured altitudes and a monotonically decreasing temperature with pressure at all longitudes. We also derive a Bond albedo of 0.18(-0.12)(+0.07) and an altitude dependence in the hot-spot offset relative to the substellar point.