R-band intensity measurements along the light curve of Type Ia supernovae discovered by the Cosmology Project (SCP) are fitted in brightness to templates allowing a free parameter the time-axis width factor w identically equal to s times (1+z). The data points are then individually aligned in the time-axis, normalized and K-corrected back to the rest frame, after which the nearly 1300 normalized intensity measurements are found to lie on a well-determined common rest-frame B-band curve which we call the "composite curve." The same procedure is applied to 18 low-redshift Calan/Tololo SNe with Z < 0.11; these nearly 300 B-band photometry points are found to lie on the composite curve equally well. The SCP search technique produces several measurements before maximum light for each supernova. We demonstrate that the linear stretch factor, s, which parameterizes the light-curve timescale appears independent of z, and applies equally well to the declining and rising parts of the light curve. In fact, the B band template that best fits this composite curve fits the individual supernova photometry data when stretched by a factor s with chi 2/DoF ~;~; 1, thus as well as any parameterization can, given the current data sets. The measurement of the data of explosion, however, is model dependent and not tightly constrained by the current data. We also demonstrate the 1 + z light-cure time-axis broadening expected from cosmological expansion. This argues strongly against alternative explanations, such as tired light, for the redshift of distant objects.