  We propose the operation of \textbf{LEvEL}, the Low-Energy Neutrino
Experiment at the LHC, a neutrino detector near the Large Hadron Collider Beam
Dump. Such a detector is capable of exploring an intense, low-energy neutrino
flux and can measure neutrino cross sections that have previously never been
observed. These cross sections can inform other future neutrino experiments,
such as those aiming to observe neutrinos from supernovae, allowing such
measurements to accomplish their fundamental physics goals. We perform detailed
simulations to determine neutrino production at the LHC beam dump, as well as
neutron and muon backgrounds. Measurements at a few to ten percent precision of
neutrino-argon charged current and neutrino-nucleus coherent scattering cross
sections are attainable with 100~ton-year and 1~ton-year exposures at LEvEL,
respectively, concurrent with the operation of the High Luminosity LHC. We also
estimate signal and backgrounds for an experiment exploiting the forward
direction of the LHC beam dump, which could measure neutrinos above 100 GeV.
