  The level density of quantum states in statistical mesoscopic systems is a
critical input for various fields of physics, including nuclear physics,
nuclear astrophysics, atomic physics and their applications. In atomic nuclei,
the level density is a fundamental measure of their complex structure at
relatively high energies. Here we present the first model-independent
measurement of the absolute partial nuclear level density for a short-lived
unstable nucleus. For this purpose, we introduce the ``Shape method'' to
extract the shape of the $\gamma$-ray strength function. Combining the Shape
method with the existing $\beta$-Oslo technique allows the extraction of the
nuclear level density without the need for theoretical input. We benchmark the
Shape method using results for the stable $^{76}$Ge nucleus, finding an
excellent agreement to previous experimental results. We apply the Shape method
to new experimental data on the short-lived $^{88}$Kr nucleus. Our method opens
the door for measurements of the nuclear level density and $\gamma$-ray
strength function far away from stability, a pivotal input required to
understand the role of exotic nuclei in forming the cosmos.
