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

Given,

$\rm a = \frac{1}{64}$ and $\rm b = \frac{1}{16}$

By substituting the values into the formula, we get,

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

$\rm or, GM = \sqrt{ \frac{1}{64} \cdot \frac{1}{16} }$

$\rm or, GM = \sqrt{ \frac{1}{64 \cdot 16}}$

$\rm or, GM = \sqrt{\frac{1}{1024}}$

$\rm or, GM = \sqrt{ \frac{1}{ 32^{2} }}$

$\rm \therefore GM = \frac{1}{32}$

Hence, the required geometric mean between the numbers $\rm \frac{1}{64}$ and $\rm \frac{1}{16}$ is $\rm \frac{1}{32}$.