The recent discovery of superconductivity in pressurized Ruddlesden-Popper (RP) nickelates has provided new perspectives on the mechanism of high-temperature superconductivity. Up to now, most experiments concentrated on the lanthanum-related RP phase, so the discovery of new superconducting RP nickelates is highly desirable to reveal their generality. Here we report that high-quality Pr4Ni3O10 single crystal is grown with an optical floating zone furnace under high oxygen pressure. High-pressure transport measurements show that the superconducting state arises above 10 GPa, and the maximum Tc reaches 39 K without saturation, significantly exceeding the value of 25-30 K of La4Ni3O10. Ultrasensitive d.c. magnetic susceptibility measurements under high pressure indicate bulk superconductivity with appreciable superconducting volume fractions. By performing in situ high-pressure synchrotron X-ray diffraction measurements at 16 K, a structural transition is found from monoclinic to tetragonal. Unlike La4Ni3O10, the electronic structure of the high-pressure phase of Pr4Ni3O10 from density functional theory exhibits a dramatic metallization of the sigma-bonding band consisting of three dz2 orbitals and van Hove singularity of coupled bands of dx2-y2 orbitals near the Fermi level, similar to the bilayer nickelate La3Ni2O7. These findings reveal some generic features of both crystal and electronic structures for high-temperature superconductivity in nickelates and multi-layer cuprates.