  Recent studies have shown that the observed colour distributions of Type Ia
SNe (SNIa) are well-described by a combination of distributions from dust and
intrinsic colour. Here we present a new forward-modeling fitting method
(Dust2Dust) to measure the parent dust and colour distributions, including
their dependence on host-galaxy mass. At each fit step, the SNIa selection
efficiency is determined from a large simulated sample that is re-weighted to
reflect the proposed distributions. We use five separate metrics to constrain
the Dust2Dust parameters: distribution of fitted light-curve colour $c$,
cosmological residual trends with $c$, cosmological residual scatter with $c$,
fitted colour-luminosity relationship $\beta_{\rm SALT2}$, and intrinsic
scatter $\sigma_{\rm int}$. Using the Pantheon+ data sample, we present results
for a Dust2Dust fit that includes 4 parameters describing intrinsic colour
variations and 8 parameters describing dust. Furthermore, we propagate the
Dust2Dust parameter uncertainties and covariance to the dark energy
equation-of-state $w$ and Hubble constant H$_0$: we find $\sigma_w = 0.005$ and
$\sigma_{\textrm{H}_0} = 0.145~$km/s/Mpc. The Dust2Dust code is publically
available.
