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

Given,

$\rm a = 81$

$\rm GM = 27$

By substituting the values into the formula, we get,

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

$\rm or, GM^{2} = ab$

$\rm or, b = \frac{GM^{2}}{a}$

$\rm or, b = \frac{ 27^{2}}{81}$

$\rm \therefore b = 9$

Hence, the required second number is 9.