  Finite-momentum Cooper pairing is an unconventional form of superconductivity
that is widely believed to require finite magnetization. Altermagnetism is an
emerging magnetic phase with highly anisotropic spin-splitting of specific
symmetries, but zero net magnetization. Here, we study Cooper pairing in
metallic altermagnets connected to conventional $s$-wave superconductors.
Remarkably, we find that the Cooper pairs induced in the altermagnets acquire a
finite centre-of-mass momentum, despite the \textit{zero} net magnetization in
the system. This anomalous Cooper-pair momentum strongly depends on the
propagation direction and exhibits unusual symmetric patterns. Furthermore, it
yields several unique features: (i) highly orientation-dependent oscillations
in the order parameter, (ii) controllable 0-$\pi$ transitions in the Josephson
supercurrent, (iii) large-oblique-angle Cooper-pair transfer trajectories in
junctions parallel with the direction where spin splitting vanishes, and (iv)
distinct Fraunhofer patterns in junctions oriented along different directions.
Finally, we discuss the implementation of our predictions in candidate
materials such as RuO$_{2}$ and KRu$_{4}$O$_{8}$.
