The Penning trap has been investigated as the basis of a small nuclear fusion reactor using a superconducting solenoid magnet. To extend this investigation, we designed, constructed, and evaluated a permanent magnet Penning trap. The device consists of a solenoid formed from an annular array of neodymium bar magnets between two iron pole pieces designed to give a uniform magnetic field in the central volume of the device. Critical to achieving the uniform solenoidal field is an iron equatorial ring supported within the annular array of magnets. A nonmagnetic titanium Penning trap with hyperbolic surfaces designed to produce a spherical potential well was mounted inside the permanent magnet assembly. The trap was fitted with a nonmagnetic hairpin filament electron source and demonstrated to produce electron trapping at the theoretically predicted magnetic fields and trap potentials. Trap potentials achievable were limited by electrical breakdown within the trap operating in constant potential mode. Efforts were made to extend the trap potentials using pulsed anode voltages, but nuclear fusion in a Penning trap has not yet been demonstrated. The design and construction of the permanent magnet solenoid and nonmagnetic trap are presented here as potentially useful also in other studies.