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.
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