If the given quadratic equation has one solution, it follows that its discriminant must be equal to $ 0 $. The discriminant of the given quadratic is given by $ m^2 - 4(m+n) $, and setting this equal to zero, we obtain another quadratic equation $ m^2 - 4m - 4n = 0 $. Since the value of $ m $ is unique, it follows that again, the discriminant of this quadratic must be equal to zero. The discriminant is now $ 4^2 - 4(-4n) = 16 + 16n = 0 $, so it follows that $ n = \boxed{-1} $.