  The cumulative emission of Axion-Like Particles (ALPs) from all past
core-collapse supernovae (SNe) would lead to a diffuse flux with energies
${\mathcal O}(50)$ MeV. We use this to constrain ALPs featuring couplings to
photons and to nucleons. ALPs coupled only to photons are produced in the SN
core via the Primakoff process, and then converted into gamma rays in the
Galactic magnetic field. We set a bound on $g_{a\gamma} \lesssim 5 \times
10^{-10}~{\rm GeV}^{-1}$ for $m_a \lesssim 10^{-11}~{\rm eV}$, using recent
measurements of the diffuse gamma-ray flux observed by the Fermi-LAT telescope.
However, if ALPs couple also with nucleons, their production rate in SN can be
considerably enhanced due to the ALPs nucleon-nucleon bremsstrahlung process.
Assuming the largest ALP-nucleon coupling phenomenologically allowed, bounds on
the diffuse gamma-ray flux lead to a much stronger $g_{a\gamma} \lesssim 6
\times 10^{-13}~{\rm GeV}^{-1}$ for the same mass range. If ALPs are heavier
than $\sim$ keV, the decay into photons becomes significant, leading again to a
diffuse gamma-ray flux. In the case of only photon coupling, we find, e.g.
$g_{a\gamma} \lesssim 5 \times 10^{-11}~{\rm GeV}^{-1}$ for $m_a \sim 5~{\rm
keV}$. Allowing for a (maximal) coupling to nucleons, the limit improves to the
level of $g_{a\gamma} \lesssim 10^{-19}~{\rm GeV}^{-1}$ for $m_a \sim 20~{\rm
MeV}$, which represents the strongest constraint to date.
