We present a measurement of the elastic differential cross section dσ(pp → pp)/dt as a function of the four-momentum-transfer squared t. The data sample corresponds to an integrated luminosity of ≈ 31 nb −1 collected with the D0 detector using dedicated Tevatron pp Collider operating conditions at √ s = 1.96 TeV and covers the range 0.26 < |t| < 1.2 GeV 2 . For |t| < 0.6 GeV 2 , dσ/dt is described by an exponential function of the form Ae −b|t| with a slope parameter b = 16.86 ± 0.10 (stat) ± 0.20 (syst) GeV −2 . A change in slope is observed at |t| ≈ 0.6 GeV 2 , followed by a more gradual |t| dependence with increasing values of |t|.