The solid-liquid phase equilibria of the Gaussian core model are determined using the GWTS algorithm, which combines equilibrium and nonequilibrium molecular dynamics simulations. This is the first reported use of the GWTS algorithm for a fluid system displaying a reentrant melting scenario. Using the GWTS algorithm, the phase envelope of the Gaussian core model can be calculated more precisely than previously possible. The results for the low-density and the high-density (reentrant melting) sides of the solid state are in good agreement with those obtained by Monte Carlo simulations in conjunction with calculations of the solid free energies. The common point on the Gaussian core envelope, where equal-density solid and liquid phases are in coexistence, could be determined with high precision.