1. **State the problem:** Find $\frac{2}{3}$ of $\frac{1}{5}$. This means multiply $\frac{2}{3}$ by $\frac{1}{5}$. 2. **Formula used:** To find a fraction of another fraction, multiply the numerators and multiply the denominators: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Apply the formula:** $$\frac{2}{3} \times \frac{1}{5} = \frac{2 \times 1}{3 \times 5} = \frac{2}{15}$$ 4. **Simplify if possible:** The numerator is 2 and denominator is 15. They have no common factors other than 1, so the fraction is already in simplest form. 5. **Final answer:** $$\boxed{\frac{2}{15}}$$ This means two-thirds of one-fifth is two-fifteenths.