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We analyze and report in detail new near-infrared (1.45-2.45 μm) observations of the Dusty S-cluster Object (DSO/G2) during its approach to the black hole at the center of the Galaxy that were carried out with the ESO Very Large Telescope/SINFONI between 2014 February and September. Before 2014 May we detect spatially compact Brγ and Paα line emission from the DSO at about 40 mas east of Sgr A*. The velocity of the source, measured from the redshifted emission, is 2700 ± 60 km s–1. No blueshifted emission above the noise level is detected at the position of Sgr A* or upstream of the presumed orbit. After May we find spatially compact Brγ blueshifted line emission from the DSO at about 30 mas west of Sgr A* at a velocity of –3320 ± 60 km s–1 and no indication for significant redshifted emission. We do not detect any significant extension of the velocity gradient across the source. We find a Brγ line FWHM of 50 ± 10 Å before and 15 ± 10 Å after the peribothron transit, i.e., no significant line broadening with respect to last year is observed. Brγ line maps show that the bulk of the line emission originates from a region of less than 20 mas diameter. This is consistent with a very compact source on an elliptical orbit with a peribothron time passage in 2014.39 ± 0.14. For the moment, the flaring activity of the black hole in the near-infrared regime has not shown any statistically significant increment. Increased accretion activity of Sgr A* may still be upcoming. We discuss details of a source model according to which the DSO is a young accreting star rather than a coreless gas and dust cloud. [ABSTRACT FROM AUTHOR]