Verification of theoretical predictions of an oscillating behavior of the fine-structure constant, $\alpha$, with cosmic time requires high precision measurements at individual redshifts, while in earlier studies the mean $\Delta\alpha/\alpha$ values averaged over wide redshift intervals were usually reported. This requirement can be met via the Single Ion Differential $\alpha$ Measurement (SIDAM) procedure. We apply SIDAM to the FeII lines associated with the damped Ly$\alpha$ system observed at $z = 1.15$ in the spectrum of HE0515–4414. The weighted mean calculated on the base of carefully selected 34 FeII pairs is $\langle \Delta\alpha/\alpha \rangle = (-0.07\,{\pm}\,0.84)\times10^{-6}$. The precision of this estimate represents a large improvement over previous measurements of $\Delta\alpha/\alpha$.