jmc

Consider the quadratic formula $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Since the quadratic has exactly one real root, then its discriminant must be 0. Thus, this gives us $$\begin{array}{rl}0& =b^2-4ac\\ \Rightarrow 0& =(4m{)}^2-4m\\ \Rightarrow 0& =16m^2-4m\\ \Rightarrow 0& =4m(4m-1).\end{array}$$This gives us the two possible values of $m$: $0$ and $\frac{1}{4}$. Since the question only asks for the positive value, our final answer is $\boxed{\frac{1}{4}}$.