This study starts with a simple question: can the vibration of plates or beams be efficiently reduced using a lightweight structure that occupies a small space? As an efficient technique to damp vibration, the concept of an acoustic black hole (ABH) is adopted with a simple modification of the geometry. The original shape of an ABH is a straight wedge-type profile with power-law thickness, with the reduction of vibration in beams or plates increasing as the length of the ABH increases. However, in real-world applications, there exists an upper bound of the length of an ABH due to space limitations. Therefore, in this study, the authors propose a curvilinear shaped ABH using the simple mathematical geometry of an Archimedean spiral, which allows a uniform gap distance between adjacent baselines of the spiral. In numerical simulations, the damping performance increases as the arc length of the Archimedean spiral increases, regardless of the curvature of the spiral in the mid- and high-frequency ranges. Adding damping material to an ABH can also strongly enhance the damping performance while not significantly increasing the weight. In addition, the radiated sound power of a spiral ABH is similar to that of a standard ABH.