jmc

Multiplying the numerator and denominator by the conjugate of the denominator, we have $$\begin{array}{rl}\frac{1-i}{2+3i}\cdot \frac{2-3i}{2-3i}& =\frac{1(2)+1(-3i)-i(2)-i(-3i)}{2(2)+2(-3i)+3i(2)-3i(3i)}\\ & =\frac{-1-5i}{13}\\ & =\boxed{-\frac{1}{13}-\frac{5}{13}i}.\end{array}$$