We present new results on the Hubble diagram of distant type Ia supernovae (SNe Ia) segregated according to the type of host galaxy. This makes it possible to check earlier evidence for a cosmological constant by explicitly comparing SNe residing in galaxies likely to contain negligible dust with the larger sample. The cosmological parameters derived from these SNe Ia hosted by presumed dust-free early-type galaxies support earlier claims for a cosmological constant, which we demonstrate at ≃ 5σ significance, and the internal extinction implied is small even for late-type systems (AB < 0.2). Furthermore, the scatter observed in the SNe Ia Hubble diagrams correlates closely with host galaxy morphology. We find this scatter is smallest for SNe Ia occurring in early-type hosts and largest for those in late-type galaxies. Moreover, SNe residing in late-type hosts appear fainter in their light-curve-width-corrected luminosity than those in early-type hosts, as expected if a modest amount of dust extinction is a contributing factor. Thus, our data demonstrate that host galaxy extinction is unlikely to systematically dim distant SNe Ia in a manner that would produce a spurious cosmological constant.