A logic problem by Christopher Boo

$\large \begin{matrix} & A & B & C \\ & & B & C \\ + & & & C \\ \hline \\ \ \\ \ \end{matrix}$

$A$ , $B$ , and $C$ in the cryptogram above are distinct digits from 1 to 9. What is the largest $\overline{ABC}$ such that the sum is the smallest possible 4-digit number?