We present results from a joint ALMA/HST study of the nearby spiral galaxy NGC.628. We combine the Hubble Space Telescope (HST) Legacy ExtraGalactic UV Survey (LEGUS) database of over 1000 stellar clusters in NGC.628 with ALMA Cycle 4 mm/submillimeter observations of the cold dust continuum that span similar to 15.kpc(2) including the nuclear region and western portions of the galaxy's disk. The resolution-1 ''.1 or approximately 50 pc at the distance of NGC.628-allows us to constrain the spatial variations in the slope of the millimeter dust continuum as a function of the ages and masses of the nearby stellar clusters. Our results indicate an excess of dust emission in the millimeter, assuming a typical cold dust model for a normal star-forming galaxy, but little correlation of the dust continuum slope with stellar cluster age or mass. For the depth and spatial coverage of these observations, we cannot substantiate the millimeter/submillimeter excess arising from the processing of dust grains by the local interstellar radiation field. We detect a bright unknown source in NGC.628 in ALMA bands 4 and 7 with no counterparts at other wavelengths from ancillary data. We speculate this is possibly a dust-obscured supernova.