The result of slicing through the center of the solid $S$ is the base described by the ellipse $\frac{x^{2}}{64}+\frac{y^{2}}{9}=1$. Cross sections of $S$ perpendicular to the $x$-axis and the elliptical base are rectangles whose height from the base is one-third its length. What is the volume of $S$ ?

Step 1 ：The volume of a solid can be calculated by integrating the area of the cross sections. In this case, the cross sections are rectangles. The length of the rectangle is the diameter of the ellipse at a given x-value, and the height is one-third of the length.

Step 2 ：The area of a rectangle is length times width, so the area of the cross section at a given x-value is \(\frac{1}{3} \times length^2\). The length of the rectangle is the diameter of the ellipse at a given x-value, which is \(2\sqrt{9-\frac{9x^2}{64}}\).

Step 3 ：Therefore, the area of the cross section at a given x-value is \(\frac{1}{3} \times (2\sqrt{9-\frac{9x^2}{64}})^2\).

Step 4 ：To find the volume of the solid, we integrate this expression from -8 to 8 (the x-values at which the ellipse intersects the x-axis).

Step 5 ：After calculating, we find that the volume of the solid is \(\boxed{128}\) cubic units.