We assess the cosmological variability of the fine-structure constant \alpha from the analysis of an ensemble of Fe ii \lambda 1608 , \lambda 2344 , \lambda 2374 , \lambda 2383 , \lambda 2587 , and \lambda 2600 absorption lines at the redshift z = 1.15 toward the QSO HE 0515–4414 by means of the standard many-multiplet ( MM ) technique and its revision based on linear regression ( RMM ) . This is the first time the MM technique is applied to exceptional high-resolution and high signal-to-noise QSO spectra recorded with the UV-Visual Echelle Spectrograph ( UVES ) at the ESO Very Large Telescope ( VLT ) . Our analysis results in \langle \Delta \alpha / \alpha \rangle _ { \mathrm { MM } } = ( 0.1 \pm 1.7 ) \cdot 10 ^ { -6 } and \langle \Delta \alpha / \alpha \rangle _ { \mathrm { RMM } } = ( -0.4 \pm 1.9 \pm 2.7 _ { \mathrm { % sys } } ) \cdot 10 ^ { -6 } , which are the most stringent bounds hitherto infered from an individual QSO absorption system . Our results support the null hypothesis \langle \Delta \alpha / \alpha \rangle = 0 at a significance level of 91 percent , whereas the support for the result \langle \Delta \alpha / \alpha \rangle = -5.7 \cdot 10 ^ { -6 } presented in former MM studies is 12 percent .