The volume \( V \) of a cube can be calculated using the formula:
\[ V = \text{side length}^3 \]
For a cube with a side length of \( 1.5 \) inches:
\[ V = (1.5)^3 \]
Calculating \( (1.5)^3 \):
\[ 1.5 \times 1.5 = 2.25 \] \[ 2.25 \times 1.5 = 3.375 \]
Thus, the volume of the cube is:
\[ V = 3.375 , \text{in}^3 \]
Rounded to the nearest thousandth, the answer is:
\[ \boxed{3.375} \]