The volume of a sphere is given by the formula (4/3)πr^3, where r is the radius of the sphere, so when dealing with ratios of diameters, the volume will follow the cube of the ratio of the diameters.

In the problem, the diameter ratio of the two new balls to the original ball is 1 : 2. Since the smaller ball has half the diameter of the original ball, the corresponding volume ratio is (1^3) : (2^3), which is 1 : 8. This means the volume of the smaller ball is one-eighth that of the original ball.

However, the ratio we want is between the smaller new ball and the original ball. As the original ball was split into two new balls, the volume of the original ball would be distributed between the two new balls. Therefore, each of these new balls, including the smaller ball, would have a portion of the original ball`s volume, equating to 1 : 9.

Option 4 stating 1 : 9 is the correct ratio of the volume of the smaller new ball to the original ball.