jmc

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 $n=\boxed{-1}$.