Rather than having the artificial situation where the surrounding gas (air)
finishes at the temperature of freezing water, why not treat the surrounding
air as a temperature reservoir. This surely is much more realistic.
To calculate the entropy change of the surroundings (whose initial and final
temperature is T_g, a temperature lower than T_0) consider a reversible
change where there is an increase of energy mL_f at a temperature T_g.
The entropy change is mL_f/T_g.
The net entropy change mL_f(1//T_g.- 1/T_0) is positive