We present new results on the Hubble diagram of distant type Ia supernovae (SNeIa) 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 SNeIa 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). 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. Our analysis is based on new Hubble Space Telescope STIS `snapshot' images and Keck-II echellette spectroscopy at the locations of the SNe, spanning the redshift range 0 < z < 0.8. Selecting from the sample discovered by the Supernova Cosmology Project (SCP), we classify the host galaxies of 39 distant SNe using the combination of STIS imaging, Keck spectroscopy and ground-based broad-band photometry. The distant data are analysed in comparison with a low-redshift sample of 25 SNe Ia re-calibrated according to the precepts of the SCP. 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 occurring in late-type galaxies. Moreover, SNe residing in late-type hosts appear ~= 0.14 +/- 0.09 mag 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. As in previous studies, these results are broadly independent of whether corrections based upon SN light-curve shapes are performed. We also use our high-redshift data set to search for morphological dependences in the SNe light curves, as are sometimes seen in low-redshift samples. No significant trends are found, possibly because the range of light-curve widths is too limited.