Verification of theoretical predictions of an oscillating behavior of the fine-structure constant α (≡ e/h̄c) with cosmic time requires high precision ∆α/α measurements at individual redshifts, while in earlier studies the mean ∆α/α values averaged over wide redshift intervals were usually reported. This requirement can be met via the single ion differential α measurement (SIDAM) procedure proposed in Levshakov et al. (2005). We apply the SIDAM to the Fe ii lines associated with the damped Lyα system observed at zabs = 1.15 in the spectrum of HE 0515–4414. The weighted mean 〈∆α/α〉 calculated on base of carefully selected 34 Fe ii pairs {λ1608, X} (X = 2344, 2374, and 2586 Å) is 〈∆α/α〉 = (−0.07 ± 0.84) × 10 (1σ C.L.). The precision of this estimate improves by a factor 2 the previous one reported for the same system by Quast et al. (2004). The obtained result represents an absolute improvement with respect to what has been done in the measurements of ∆α/α .