  We report on precise measurements of absolute nonlinear ionization
probabilities obtained by exposing optically trapped ultracold rubidium atoms
to the field of an ultrashort laser pulse in the intensity range of $1 \times
10^{11}$ to $4 \times 10^{13}$ W/cm$^2$. The experimental data are in perfect
agreement with ab-initio theory, based on solving the time-dependent
Schr\"odinger equation without any free parameters. Ultracold targets allow to
retrieve absolute probabilities since ionized atoms become apparent as a local
vacancy imprinted into the target density, which is recorded simultaneously. We
study the strong-field response of $^{87}$Rb atoms at two different wavelengths
representing non-resonant and resonant processes in the demanding regime where
the Keldysh parameter is close to unity.
