  Context. In the Milky Way, most globular clusters are highly conspicuous
objects that were found centuries ago. However, a few dozen of them are faint,
sparsely populated systems that were identified largely during the second half
of the past century. One of the faintest is ESO 37-1 (E 3) and as such it
remains poorly studied, with no spectroscopic observations published so far,
although it was discovered in 1976.
  Aims. We investigate the globular cluster E 3 in an attempt to better
constrain its fundamental parameters. Spectroscopy of stars in the field of E 3
is shown here for the first time.
  Methods. Deep, precise VI CCD photometry of E 3 down to V=26 mag is presented
and analysed. Low-resolution, medium signal-to-noise ratio spectra of nine
candidate members are studied to derive radial velocity and metallicity. Proper
motions from the UCAC4 catalogue are used to explore the kinematics of the
bright members of E 3.
  Results. Isochrone fitting indicates that E 3 is probably very old, with an
age of about 13 Gyr; its distance from the Sun is nearly 10 kpc. It is also
somewhat metal rich with [Fe/H]=-0.7. Regarding its kinematics, our tentative
estimate for the proper motions is (-7.0+/-0.8, 3.5+/-0.3) mas/yr (or a
tangential velocity of 382+/-79 km/s) and for the radial velocity is 45+/-5
km/s, in the solar rest frame.
  Conclusions. E 3 is one of the most intriguing globular clusters in the
Galaxy. Having an old age and being metal rich is clearly a peculiar
combination, only seen in a handful of objects like the far more conspicuous
NGC 104 (47 Tucanae). In addition, its low luminosity and sparse population
make it a unique template for the study of the final evolutionary phases in the
life of a star cluster. Unfortunately, E 3 is among the most elusive and
challenging known globular clusters because field contamination severely
hampers spectroscopic studies.
