The given problem states that when we divide a fraction by the difference between 1/2 and 1/6, we obtain the result of 2/3. Let's break down the solution step by step:

1. Understanding the problem:
- We are given a fraction that needs to be divided by the difference of two fractions.
- The result of this division is equal to 2/3.

2. Identifying the fractions:
- The difference between 1/2 and 1/6 can be calculated as follows:
1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3.
- Therefore, the difference between 1/2 and 1/6 is 1/3.

3. Setting up the equation:
- Let's assume the fraction we need to divide is represented by 'x'.
- We can now set up the equation as follows: x ÷ (1/3) = 2/3.

4. Solving the equation:
- To divide by a fraction, we multiply by its reciprocal. Hence, the equation becomes: x × (3/1) = 2/3.
- Simplifying, we have: 3x = 2/3.
- Multiplying both sides of the equation by 1/3, we get: x = (2/3) × (1/3).
- Multiplying the numerators and denominators, we have: x = 2/9.

5. Finding the fraction:
- The solution to the equation is x = 2/9.
- Therefore, the fraction we started with is 2/9.

In summary, when we divide the fraction 2/9 by the difference between 1/2 and 1/6 (which is 1/3), we obtain the result of 2/3.

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