Proton-proton elastic scattering has been measured by the TOTEM experiment at the CERN Large Hadron Collider at √ s = 7 TeV in dedicated runs with the Roman Pot detectors placed as close as seven times the transverse beam size (σ beam ) from the outgoing beams. After careful study of the accelerator optics and the detector alignment, |t| , the square of four-momentum transferred in the elastic scattering process, has been determined with an uncertainty of δt = 0.1 GeV |t|. In this letter, first results of the differential cross section are presented covering a |t|-range from 0.36 to 2.5 GeV 2 . The differential cross-section in the range 0.36 < |t| < 0.47 GeV 2 is described by an exponential with a slope parameter B = (23.6 ± 0.5 stat ± 0.4 syst ) GeV −2 , followed by a significant diffractive minimum at |t| = (0.53 ± 0.01 stat ± 0.01 syst ) GeV 2 . For |t|-values larger than ∼ 1.5 GeV 2 , the cross-section exhibits a power law behaviour with an exponent of -7.8 ± 0.3 stat ± 0.1 syst . When compared to predictions based on the different available models, the data show a strong discriminative power despite the small t-range covered.