It is difficult to efficiently harvest the energy of weak and broadband vibrations existing in the environment. In this study, a piezoelectric vibration energy harvester incorporating a lever structure and bi-stability is proposed. Under base excitation, the lever can amplify the displacement and make the piezoelectric beam execute a bi-stable snap-through motion easily, thereby producing large electric outputs. The electromechanical equations are derived, and corresponding simulations are carried out. The analyses on potential energy indicate that the harvester can be symmetrically and asymmetrically bi-stable under gravity. Then the harvester's prototype is fabricated and the validation experiments are carried out. The frequency-sweeping experiment shows that the harvester owns a wide frequency band. The experiment for random excitation proves that the harvester could execute bi-stable vibration and produce relatively high output under broadband and weak excitations. For a random excitation with a power spectral density (PSD) of 0.035 g2/Hz, the harvester can generate an output power with a root mean square (RMS) of 29.41 μW.