Using precise full-sky observations from Planck, and applying several methods of component separation, we identify and characterise the emission from the Galactic "haze" at microwave wavelengths. The haze is a distinct component of diffuse Galactic emission, roughly centered on the Galactic centre, and extends to | b | ~ 35-50° in Galactic latitude and | l | ~ 15-20° in longitude. By combining the Planck data with observations from the Wilkinson Microwave Anisotropy Probe, we were able to determine the spectrum of this emission to high accuracy, unhindered by the strong systematic biases present in previous analyses. The derived spectrum is consistent with power-law emission with a spectral index of -2.56 ± 0.05, thus excluding free-free emission as the source and instead favouring hard-spectrum synchrotron radiation from an electron population with a spectrum (number density per energy) dN/dE ∝ E-2.1. At Galactic latitudes | b | < 30°, the microwave haze morphology is consistent with that of the Fermi gamma-ray "haze" or "bubbles", while at b ~ -50° we have identified an edge in the microwave haze that is spatially coincident with the edge in the gamma-ray bubbles. Taken together, this indicates that we have a multi-wavelength view of a distinct component of our Galaxy. Given both the very hard spectrum and the extended nature of the emission, it is highly unlikely that the haze electrons result from supernova shocks in the Galactic disk. Instead, a new astrophysical mechanism for cosmic-ray acceleration in the inner Galaxy is implied.