We theoretically show that a magnet can be stably levitated on top of a punctured superconductor sheet in the Meissner state without applying any external field. The trapping potential created by such induced-only superconducting currents is characterized for magnetic spheres ranging from tens of nanometers to tens of millimeters. Such a diamagnetically levitated magnet is predicted to be extremely well isolated from the environment. We propose to use it as an ultrasensitive force and inertial sensor. A magneto-mechanical readout of its displacement can be performed by using superconducting quantum interference devices. An analysis using current technology shows that force and acceleration sensitivities on the order of 10(-23) N/root Hz (for a 100-nm magnet) and 10(-14) g/root Hz (for a 10-mm magnet) might be within reach in a cryogenic environment. Such remarkable sensitivities, both in force and acceleration, can be used for a variety of purposes, from designing ultrasensitive inertial sensors for technological applications (e.g., gravimetry, avionics, and space industry), to scientific investigations on measuring Casimir forces of magnetic origin and gravitational physics.