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

Given,

$\rm a = 256$ and $\rm b = 64$

By substituting the values into the formula, we get,

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

$\rm or, GM = \sqrt{ 256 \cdot 64 }$

$\rm or, GM = \sqrt{16384}$

$\rm or, GM = \sqrt{ 128^{2} }$

$\rm \therefore GM = 128$

Hence, the required geometric mean between the numbers $\rm 256$ and $\rm 64$ is $\rm 128$.