To divide fractions, you multiply by the reciprocal of the divisor. The reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
So,
\[ \frac{1}{9} \div \frac{2}{3} = \frac{1}{9} \times \frac{3}{2}. \]
Now, multiply the fractions:
\[ \frac{1 \times 3}{9 \times 2} = \frac{3}{18}. \]
Next, simplify \(\frac{3}{18}\):
\[ \frac{3 \div 3}{18 \div 3} = \frac{1}{6}. \]
Thus,
\[ \frac{1}{9} \div \frac{2}{3} = \frac{1}{6}. \]