A recently derived numerical algorithm for one-dimensional one-phase Stefan problems is extended for the purpose of two-phase moving boundary problems in which the second phase first appears only after a finite delay time; this can occur if the phase change is caused by a heat-flux boundary condition. In tandem with the Keller box finite-difference scheme, the so-called boundary immobilization method is used. An important component of the work is the use of variable transformations that must be built into the numerical algorithm to resolve the boundary-condition discontinuity that is associated with the onset of phase change. This allows the delay time until solidification begins to be determined, and gives second-order accuracy in both time and space.