To find the volume of the sphere, we first need to determine the dimensions of the cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone. According to the problem, the volume of the cone is \( 253 \pi , \text{cm}^3 \). Thus, we can set up the equation:
\[ \frac{1}{3} \pi r^2 h = 253 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{1}{3} r^2 h = 253 \]
Multiplying both sides by 3 gives:
\[ r^2 h = 759 \quad \text{(1)} \]
We are also told that the height of the cone \( h \) is equal to the diameter of the sphere. Since the radius of the sphere is the same as that of the cone, the diameter of the sphere \( d \) would be:
\[ d = 2r \]
Thus, we can express the height of the cone in terms of the radius:
\[ h = 2r \quad \text{(2)} \]
Now we can substitute equation (2) into equation (1):
\[ r^2 (2r) = 759 \]
This simplifies to:
\[ 2r^3 = 759 \]
Dividing both sides by 2:
\[ r^3 = \frac{759}{2} = 379.5 \quad \text{(3)} \]
Next, we need to find the volume of the sphere, which is given by the formula:
\[ V_s = \frac{4}{3} \pi r^3 \]
Using equation (3), we can substitute for \( r^3 \):
\[ V_s = \frac{4}{3} \pi \left( \frac{759}{2} \right) \]
Calculating this gives:
\[ V_s = \frac{4 \cdot 759}{6} \pi = \frac{3036}{6} \pi = 506 \pi , \text{cm}^3 \]
Therefore, the volume of the sphere is:
\[ \boxed{506 \pi , \text{cm}^3} \]