MNRAS; post-referee version. 21 pages, 10 figures including 4 colour figures. Preprint also available at http://supernova.lbl.gov/(Abridged) 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 supports earlier claims for a cosmological constant, which we demonstrate at 5 sigma significance, and the internal extinction implied is small even for late-type systems (A_B<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. We classify the host galaxies of 39 distant SNe discovered by the Supernova Cosmology Project (SCP) using the combination of HST STIS imaging, Keck ESI spectroscopy and ground-based broad-band photometry. We compare with a low-redshift sample of 25 SNe Ia. The scatter observed in the SNe Ia Hubble diagrams correlates closely with host galaxy morphology. We find the scatter in the SNe Ia Hubble diagram is smallest for SNe occurring in early-type hosts and largest for those occurring in late-type galaxies. Moreover, SNe residing in early-type hosts appear only ~0.14+/-0.09 mag brighter in their light-curve-width-corrected luminosity than those in late-type hosts, implying only a modest amount of dust extinction even in the late-type systems
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