If the two numbers are a and b, then the geometric mean between them is given by $\rm GM = \sqrt{ab}$.

Given,

$\rm a = 3$ and $\rm b = 12$

By substituting the values into the formula, we get,

$\rm or, GM = \sqrt{ab}$

$\rm or, GM = \sqrt{ 3 \cdot 12 }$

$\rm or, GM = \sqrt{36}$

$\rm or, GM = \sqrt{ 6^{2} }$

$\rm \therefore GM = 6$

Hence, the required geometric mean between the numbers $\rm 3$ and $\rm 12$ is $\rm 6$.