The main question addressed in this study is whether particles with relativistic energy can be injected directly into the inner magnetosphere by very near-Earth reconnection during magnetic storms. We study a sequence of events observed in the solar wind, in the magnetotail, at geosynchronous orbit (GEO), in the inner magnetosphere, and in the ionosphere during the main phase of a geomagnetic storm on 16 June 2012–17 June 2012. The storm was caused by a magnetic cloud with a high dynamic pressure and a strong southward IMF lasting about 10 hr. These conditions caused an extreme compression of the magnetosphere (SymH reached ∼150 nT) and an enhancement of the lobe magnetic field strength to ∼90 nT at R ∼ 10 Earth radii (R_E). We focus on an hour-long interval between 1050 and 1150 UT on 17 June 2012 when the Time History of Events and Macroscale Interactions during Substorms (THEMIS) satellites were at apogee at geocentric distances R ≈ 12 R_E near midnight. At that time a thin current sheet formed between cis-GEO distances and THEMIS. This thin current sheet reconnected between GEO and THEMIS. A strong lobe magnetic field enabled ion and electron energization to energies E ≈ 600 keV. Fluxes of high energy (up to relativistic) particles at GEO increased within ≈20 s after reconnection onset detected by THEMIS. Fluxes of relativistic electrons at L ∼ 4 in the morning sector increased within about 600 s after reconnection onset. We interpret these observations as signatures of direct injection of reconnected particles into the inner magnetosphere.