  Globular clusters contain vast repositories of metal-poor stars that
represent some of the oldest stellar generations in the Universe. The
archaeological footprint of early Galactic evolution may be retained in the
measurable properties of globular clusters, such as their ages, mass functions
and chemical abundances. Until recently, all photometric studies of globular
clusters were restricted to stellar members. Now, the sensitivity of JWST can
extend this analysis to the substellar regime. If detected in sufficient
numbers, brown dwarf members can provide tight constraints on the properties of
their parent population. We present SANDee - a new grid of stellar models that
accurately represent the color-magnitude diagrams of globular clusters across
the hydrogen-burning limit at a wide range of metallicities. Using JWST NIRCam
photometry and the new models, we identify three brown dwarfs in the globular
cluster NGC 6397 with effective temperatures of 1300-1800 K, confirmed by both
proper motion and model fitting. We use the observed luminosities of discovered
brown dwarfs to obtain the first age estimate of a globular cluster from its
substellar cooling sequence: 13.4 +/- 3.3 Gyr. We also derive the local mass
function of the cluster across the hydrogen-burning limit and find it to be
top-heavy, suggesting extensive dynamical evolution. We expect that the
constraints on both age and mass function of NGC 6397 derived in this work can
be greatly improved by a second epoch of NIRCam imaging in the same field.
