We present a new, deep (V ~ 26) study of the Galactic globular cluster NGC 2419 based on B,V,I time-series CCD photometry over about 10 years and extending beyond the cluster published tidal radius. We have identified 101 variable stars of which 60 are new discoveries, doubling the known RR Lyrae stars and including 12 SX Phoenicis stars. The average period of the RR Lyrae stars (<Pab>=0.662 d, and <Pc>=0.366 d, for fundamental-mode -RRab- and first-overtone pulsators, respectively), and the position in the period-amplitude diagram both confirm that NGC 2419 is an Oosterhoff II cluster. The average apparent magnitude of the RR Lyrae stars is <V>=20.31 +/- 0.01 (sigma=0.06, 67 stars) and leads to the distance modulus (m-M)o=19.60 +/- 0.05. The Color-Magnitude Diagram, reaching about 2.6 mag below the cluster turn-off, does not show clear evidence of multiple stellar populations. Cluster stars are found until r~ 10.5, and possibly as far as r~15, suggesting that the literature tidal radius might be underestimated. No extra-tidal structures are clearly detected in the data. NGC 2419 has many blue stragglers and a well populated horizontal branch extending from the RR Lyrae stars down to an extremely blue tail ending with the blue-hook, for the first time recognized in this cluster. The red giant branch is narrow ruling out significant metallicity spreads. Our results seem to disfavor the interpretation of NGC 2419 as either having an extragalactic origin or being the relict of a dwarf galaxy tidally disrupted by the Milky Way.
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