Recent studies have shown that the observed colour distributions of Type Ia SNe (SNIa) are welldescribed 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 lightcurve colour c, cosmological residual trends with c, cosmological residual scatter with c, fitted colourluminosity relationship β SALT2 , and intrinsic scatter σ 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 σw = 0.005 and σH 0 = 0.145 km/s/Mpc. The Dust2Dust code is publically available.