jmc

By the formula for the total number of positive divisors, only natural numbers in the form $p^2$ for some prime $p$ have exactly three positive divisors. Thus we must count the number of primes between 1 and $\sqrt{1000}$ (the squares of these primes are all the natural numbers less than 1000 that have exactly three positive divisors). There are $\boxed{11}$ such primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31.