Let's break down the expression step by step:

1. **Simplify the expression inside the parentheses:**
\[
\left(\frac{5}{9} - \frac{2}{3}\right)
\]

To subtract these fractions, we need a common denominator. The least common multiple of 9 and 3 is 9. Convert \(\frac{2}{3}\) to a fraction with a denominator of 9:
\[
\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}
\]

Now we can subtract:
\[
\frac{5}{9} - \frac{6}{9} = \frac{5 - 6}{9} = \frac{-1}{9}
\]

2. **Square the result:**
\[
\left(\frac{-1}{9}\right)^2 = \frac{(-1)^2}{9^2} = \frac{1}{81}
\]

3. **Divide by \(\frac{4}{27}\):**
\[
\frac{\frac{1}{81}}{\frac{4}{27}}
\]

Dividing by a fraction is the same as multiplying by its reciprocal:
\[
\frac{1}{81} \times \frac{27}{4} = \frac{1 \times 27}{81 \times 4} = \frac{27}{324}
\]

4. **Simplify the fraction:**
\[
\frac{27}{324}
\]

Both the numerator and the denominator can be divided by 27:
\[
\frac{27 \div 27}{324 \div 27} = \frac{1}{12}
\]

So, the simplified result of the expression \(1 - \left(\frac{5}{9} - \frac{2}{3}\right)^2 : \frac{4}{27}\) is:
\[
\frac{1}{12}
\]