  In this Letter, we report the first measurement of the inelastic cross
section for antideuteron-nucleus interactions at low particle momenta, covering
a range of $0.3 \leq p < 4$ GeV/$c$. The measurement is carried out using p-Pb
collisions at a center-of-mass energy per nucleon-nucleon pair of
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV, recorded with the ALICE detector at the CERN
LHC and utilizing the detector material as an absorber for antideuterons and
antiprotons. The extracted raw primary antiparticle-to-particle ratios are
compared to the results from detailed ALICE simulations based on the GEANT4
toolkit for the propagation of antiparticles through the detector material. The
analysis of the raw primary (anti)proton spectra serves as a benchmark for this
study, since their hadronic interaction cross sections are well constrained
experimentally. The first measurement of the inelastic cross section for
antideuteron-nucleus interactions averaged over the ALICE detector material
with atomic mass numbers $\langle A \rangle$ = 17.4 and 31.8 is obtained. The
measured inelastic cross section points to a possible excess with respect to
the Glauber model parameterization used in GEANT4 in the lowest momentum
interval of $0.3 \leq p < 0.47$ GeV/$c$ up to a factor 2.1. This result is
relevant for the understanding of antimatter propagation and the contributions
to antinuclei production from cosmic ray interactions within the interstellar
medium. In addition, the momentum range covered by this measurement is of
particular importance to evaluate signal predictions for indirect dark-matter
searches.
