【Analysis】 $2 m$ is still a three-digit number and the sum of its digits is $6 \Rightarrow 2 m$ (even)
Possible values are $600,510,420,402,330,312,240,222,204$, corresponding $m$ values are $300,255,210,201,165,156,120,111,102$, verifying with the sum of digits of $m$ being 12, only $255,165,156$ are valid, a total of 3.
13. If the three-digit number $m$ simultaneously satisfies the following conditions: (1) the sum of the digits of $m$ is 12; (2) $2m$ is still a three-digit number, and the sum of its digits is 6, then the number of such three-digit numbers $m$ is ( ).