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

Given,

$\rm a = 25$ and $\rm b = 625$

By substituting the values into the formula, we get,

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

$\rm or, GM = \sqrt{ 25 \cdot 625 }$

$\rm or, GM = \sqrt{15625}$

$\rm or, GM = \sqrt{ 125^{2} }$

$\rm \therefore GM = 125$

Hence, the required geometric mean between the numbers $\rm 25$ and $\rm 625$ is $\rm 125$.