To solve the given expression \(\left(\frac{1}{2} - \frac{1}{6}\right)^3\), follow these steps:

1. Simplify the expression inside the parentheses:
\[
\frac{1}{2} - \frac{1}{6}
\]

To subtract these fractions, find a common denominator. The least common denominator of 2 and 6 is 6.

\[
\frac{1}{2} = \frac{3}{6}
\]

So,
\[
\frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
\]

2. Now, raise the simplified fraction to the power of 3:
\[
\left(\frac{1}{3}\right)^3
\]

3. Calculate the power:
\[
\left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3} = \frac{1}{27}
\]

Therefore, the result of the expression \(\left(\frac{1}{2} - \frac{1}{6}\right)^3\) is \(\frac{1}{27}\).