We construct an approximate analytic expression for the growth of the neck between two sintering roughened crystals in contact with a melt. The result combines recent work on early phase sintering [R.S. Farr, M.J. Izzard, Phys. Rev. E 77 (2008) 041608], and a numerical method presented here, based on Legendre polynomials. The expression gives an estimate which covers all times from very early to near equilibrium, and includes the effect of a grain boundary between the two crystals. The predictions are then compared to experiments on sintering of ice crystals in a sucrose solution.