Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has.  Compute the least possible value of $\left| A - B \right|$.
1. First, we need to determine the number of positive divisors of 2013. The prime factorization of 2013 is:
   \[
   2013 = 3 \times 11 \times 61
   \]
   Each exponent in the prime factorization is 1. The formula for the number of divisors of a number \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is:
   \[
   (e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
   \]
   Applying this formula to 2013, we get:
   \[
   (1+1)(1+1)(1+1) = 2 \times 2 \times 2 = 8
   \]
   Therefore, 2013 has 8 positive divisors.

2. We need to find distinct positive integers \( A \) and \( B \) such that both have exactly 8 positive divisors and compute the least possible value of \( |A - B| \).

3. To have exactly 8 divisors, a number can be of the form \( p^7 \) (where \( p \) is a prime), \( p_1^3 p_2 \) (where \( p_1 \) and \( p_2 \) are distinct primes), or \( p_1 p_2 p_3 \) (where \( p_1, p_2, p_3 \) are distinct primes).

4. We will consider the form \( p_1 p_2 p_3 \) since it is more likely to yield smaller values for \( A \) and \( B \). Let's choose small primes for \( p_1, p_2, p_3 \).

5. Consider \( A = 3 \times 5 \times 7 = 105 \) and \( B = 2^3 \times 13 = 104 \). Both numbers have 8 divisors:
   - For \( 105 = 3 \times 5 \times 7 \):
     \[
     (1+1)(1+1)(1+1) = 2 \times 2 \times 2 = 8
     \]
   - For \( 104 = 2^3 \times 13 \):
     \[
     (3+1)(1+1) = 4 \times 2 = 8
     \]

6. Calculate \( |A - B| \):
   \[
   |105 - 104| = 1
   \]

Conclusion:
The least possible value of \( |A - B| \) is \( 1 \).

The final answer is \( \boxed{1} \).