To find the volume of the sphere, we start by using the information about the cone. The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone. We are given that the volume of the cone is \( \frac{25}{3} \pi \) cm³.
We can set the volume of the cone equal to the expression above:
\[ \frac{1}{3} \pi r^2 h = \frac{25}{3} \pi \]
Since both sides have \( \frac{1}{3} \pi \), we can simplify by multiplying both sides by 3 and dividing by \( \pi \):
\[ r^2 h = 25 \]
Next, we know that the height of the cone \( h \) is equal to the diameter of the sphere. Since the diameter of the sphere is given by \( 2r \) (where \( r \) is the radius of the sphere), we have:
\[ h = 2r \]
We can substitute \( h \) into the equation \( r^2 h = 25 \):
\[ r^2 (2r) = 25 \]
This simplifies to:
\[ 2r^3 = 25 \quad \Rightarrow \quad r^3 = \frac{25}{2} \quad \Rightarrow \quad r = \left(\frac{25}{2}\right)^{1/3} \]
Now we can find the volume of the sphere \( V_s \) using the formula for the volume of a sphere:
\[ V_s = \frac{4}{3} \pi r^3 \]
Substituting \( r^3 \) into this formula:
\[ V_s = \frac{4}{3} \pi \left(\frac{25}{2}\right) = \frac{100}{6} \pi = \frac{50}{3} \pi , \text{cm}^3 \]
Thus, the volume of the sphere is:
\[ \boxed{\frac{50}{3} \pi , \text{cm}^3} \]