We note that the sum of the digits of 1000 is not 3. Every other positive integer in the given range has two or three digits. For the sum of the digits of an integer to be 3, no digit can be greater than 3. If a two-digit integer has sum of digits equal to 3, then its tens digit is 1, 2, or 3. The possible integers are 12, 21, and 30. If a three-digit integer has sum of digits equal to 3, then its hundreds digit is 1, 2, or 3. If the hundreds digit is 3, then the units and tens digits add to 0, so must be each 0. The integer must thus be 300. If the hundreds digit is 2, then the units and tens digits add to 1, so must be 1 and 0 or 0 and 1. The possible integers are 210 and 201. If the hundreds digit is 1, then the units and tens digits add to 2, so must be 2 and 0, or 1 and 1, or 0 and 2, giving possible integers 120, 111, and 102. Overall, there are 9 such positive integers.