The volume of water displaced by the sphere equals the sphere's volume: \[ \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (3)^3 = 36 \pi \, \text{cm}^3. \]
This displaced volume increases the water level in the cylindrical vessel, which has an area of \[ \pi r^2 = \pi (4)^2 = 16 \pi \, \text{cm}^2. \]
The water level rises by \[ \frac{\text{volume displaced}}{\text{area of the base}} = \frac{36\pi}{16\pi} = \frac{9}{4} \, \text{cm}. \]