We study the melting of a double stranded DNA in the presence of stretching forces, via 3D Monte-Carlo simulations, exactly solvable models and heuristic arguments. The resulting force-temperature phase diagram is dramatically different for the cases where the force is applied to only one strand or to both. Different assumptions on the monomer size of single and double stranded DNA lead to opposite conclusions as to whether DNA melts or not as it overstretches.