  HP$\,$1 is an $\alpha$-enhanced and moderately metal-poor bulge globular
cluster with a blue horizontal branch. These combined characteristics make it a
probable relic of the early star formation in the innermost Galactic regions.
Here we present a detailed analysis of a deep near-infrared (NIR) photometry of
HP$\,$1 obtained with the NIR GSAOI+GeMS camera at the Gemini-South telescope.
$J$ and $K_{\rm S}$ images were collected with an exquisite spatial resolution
(FWHM $\sim 0.1$ arcsec), reaching stars at two magnitudes below the MSTO. We
combine our GSAOI data with archival F606W-filter $HST$ ACS/WFC images to
compute relative proper motions and select bona fide cluster members. Results
from statistical isochrone fits in the NIR and optical-NIR colour-magnitude
diagrams indicate an age of $12.8^{+0.9}_{-0.8}$ Gyr, confirming that HP$\,$1
is one of the oldest clusters in the Milky Way. The same fits also provide
apparent distance moduli in the $K_{\rm S}$ and $V$ filters in very good
agreement with the ones from 11 RR Lyrae stars. By subtracting the extinction
in each filter, we recover a heliocentric distance of $6.59^{+0.17}_{-0.15}$
kpc. Furthermore, we refine the orbit of HP$\,$1 using this accurate distance
and update and accurate radial velocities (from high resolution spectroscopy)
and absolute proper motions (from Gaia DR2), reaching mean perigalactic and
apogalactic distances of $\sim$0.12 and $\sim$3 kpc respectively.
