  We present the first high-resolution Atacama Large Millimeter/Submillimeter
Array (ALMA) observations of a sunspot at wavelengths of 1.3 mm and 3 mm,
obtained during the solar ALMA Science Verification campaign in 2015, and
compare them with the predictions of semi-empirical sunspot umbral/penumbral
atmosphere models. For the first time millimeter observations of sunspots have
resolved umbral/penumbral brightness structure at the chromospheric heights,
where the emission at these wavelengths is formed. We find that the sunspot
umbra exhibits a radically different appearance at 1.3 mm and 3 mm, whereas the
penumbral brightness structure is similar at the two wavelengths. The inner
part of the umbra is ~600 K brighter than the surrounding quiet Sun (QS) at 3
mm and is ~700 K cooler than the QS at 1.3 mm, being the coolest part of
sunspot at this wavelength. On average, the brightness of the penumbra at 3 mm
is comparable to the QS brightness, while at 1.3 mm it is ~1000 K brighter than
the QS. Penumbral brightness increases towards the outer boundary in both ALMA
bands. Among the tested umbral models, that of Severino et al. (1994) provides
the best fit to the observational data, including both the ALMA data analyzed
in this study and data from earlier works. No penumbral model amongst those
considered here gives a satisfactory fit to the currently available
measurements. ALMA observations at multiple mm wavelengths can be used for
testing existing sunspot models, and serve as an important input to constrain
new empirical models.
