We perform an exact model calculation of the unbinding transition of a one-dimensional interface from a substrate on the approach of two-phase coexistence. This transition is modelled in a solid-on-solid (SOS) model with discrete columns which have a partition function calculated for a finite substrate area A. In the limits of large A and a small distance mod Delta mu mod to coexistence, the partition function becomes a function of the the scaling variable x= mod Delta mu mod 23/A. The interface-to-substrate distance l diverges as l alpha mod Delta mu mod - beta , with beta =-1/3 if the two limits are taken at constant or diverging x. If x vanishes, a finite-size regime appears in which beta =-1. A related interface model in continuous space was studied earlier by Kroll and Gompper (1989) and our scaling functions are identical to theirs.