The question of whether Cosmic Microwave Background (CMB) temperature and
polarization data from Planck favor a spatially closed Universe with curvature
parameter $\Omega_K<0$ has been the subject of recent intense discussions.
Attempts to break the geometrical degeneracy combining Planck data with
external datasets such as Baryon Acoustic Oscillation (BAO) measurements all
point towards a spatially flat Universe, at the cost of significant tensions
with Planck, which make the resulting dataset combination problematic. Settling
this issue would require identifying a dataset which can break the geometrical
degeneracy while not incurring in these tensions. In this work we argue that
cosmic chronometers (CC), measurements of the expansion rate $H(z)$ from the
relative ages of massive early-type passively evolving galaxies, are the
dataset we are after. Furthermore, CC come with the additional advantage of
being virtually free of cosmological model assumptions. Combining Planck 2018
CMB temperature and polarization data with the latest compilation of CC
measurements, we break the geometrical degeneracy and find $\Omega_K=-0.0054
\pm 0.0055$, consistent with a spatially flat Universe and competitive with the
Planck+BAO constraint. After discussing our results in light of the oldest
objects in the Universe, we assess their stability against against minimal
parameter space extensions and CC systematics, finding them to be stable
against both. We find no substantial tension between Planck and CC data within
a non-flat Universe, making the resulting combination reliable. Our results
therefore allow us to assert with confidence that the Universe is indeed
spatially flat to the ${\cal O}(10^{-2})$ level, a finding which might possibly
settle the ongoing spatial curvature debate, and lends even more support to the
already very successful inflationary paradigm.