  The recent discovery of thousands of ultra diffuse galaxies (UDGs) in nearby
galaxy clusters has opened a new window into the process of galaxy formation
and evolution. Several scenarios have been proposed to explain the formation
history of UDGs, and their ability to survive in the harsh cluster
environments. A key requirement to distinguish between these scenarios is a
measurement of their halo masses which, due to their low surface brightnesses,
has proven difficult if one relies on stellar tracers of the potential. We
exploit weak gravitational lensing, a technique that does not depend on these
baryonic tracers, to measure the average subhalo mass of 784 UDGs selected in
18 clusters at $z\leq0.09$. Our sample of UDGs has a median stellar mass
$\langle m_\star\rangle=2\times10^8\,\mathrm{M}_\odot$ and a median effective
radius $\langle r_\mathrm{eff}\rangle=2.8$ kpc. We constrain the average mass
of subhaloes within 30 kpc to $\log
m_\mathrm{UDG}(r<30\,\mathrm{kpc})/\mathrm{M}_\odot\leq10.99$ at 95 per cent
credibility, implying an effective virial mass $\log
m_{200}/\mathrm{M}_\odot\leq11.80$, and a lower limit on the stellar mass
fraction within 10 kpc of 1.0 per cent. Such mass is consistent with a simple
extrapolation of the subhalo-to-stellar mass relation of typical satellite
galaxies in massive clusters. However, our analysis is not sensitive to scatter
about this mean mass; the possibility remains that extreme UDGs reside in
haloes as massive as the Milky Way.
