The supermassive black hole at the Galactic center, Sgr A*, shows frequent radiation outbursts, often called 'flares'. In the near-infrared some of these flares were reported as showing intrinsic quasi-periodicities. The flux peaks associated with the quasi-periodic behavior were found to be highly polarized. The aim of this work is to present new evidence to support previous findings of the properties of the polarized radiation from Sgr A* and to again provide strong support for the quasi-periodicity of 18+/- 3 min reported earlier. Observations were carried out at the European Southern Observatory's Very Large Telescope on Paranal, Chile. We used the NAOS/CONICA adaptive optics/near-infrared camera instrument. By fitting the polarimetric lightcurves with a hot-spot model, we addressed the question of whether the data are consistent with this model. To fit the observed data we used a general relativistic ray-tracing code in combination with a simple hot-spot/ring model. We report on new polarization measurements of a K-band flare from the supermassive black hole at the Galactic center. The data provide very strong support for a quasi-periodicity of 15.5+/-2 min. The mean polarization of the flare is consistent with the direction of the electric field vector that was reported in previous observations. The data can be modeled successfully with a combined blob/ring model. The inclination i of the blob orbit must be i > 20 DEG on a 3 sigma level, and the dimensionless spin parameter of the black hole is derived to be a\star > 0.5.