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HP$\,$1 is an $\alpha$-enhanced and moderately metal-poor bulge globularcluster with a blue horizontal branch. These combined characteristics make it aprobable relic of the early star formation in the innermost Galactic regions.Here we present a detailed analysis of a deep near-infrared (NIR) photometry ofHP$\,$1 obtained with the NIR GSAOI+GeMS camera at the Gemini-South telescope.$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. Wecombine our GSAOI data with archival F606W-filter $HST$ ACS/WFC images tocompute relative proper motions and select bona fide cluster members. Resultsfrom statistical isochrone fits in the NIR and optical-NIR colour-magnitudediagrams indicate an age of $12.8^{+0.9}_{-0.8}$ Gyr, confirming that HP$\,$1is one of the oldest clusters in the Milky Way. The same fits also provideapparent distance moduli in the $K_{\rm S}$ and $V$ filters in very goodagreement with the ones from 11 RR Lyrae stars. By subtracting the extinctionin 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 distanceand update and accurate radial velocities (from high resolution spectroscopy)and absolute proper motions (from Gaia DR2), reaching mean perigalactic andapogalactic distances of $\sim$0.12 and $\sim$3 kpc respectively.

A Deep View of a Fossil Relic in the Galactic Bulge: The Globular Cluster HP$\,$1