Now assuming that you have the actual correlation:
Since two of the cells can be assigned arbitrarily within the constraints of the marginals the joint distribution has two degrees of freedom.
The correlation gives you one equation in the two variables that represent the degrees of freedom.
Usually a single equation in two real variables does not have a unique solution, and that is very likely the case here (it is possible that there is a unique solution but you won't know untill you so the calculations).
You almost certainly will need an additional constraint to get a unique solution.