This Letter describes the measurement of elliptic flow of charged particles in lead-lead collisions at root s(NN) = 2.76 TeV using the ATLAS detector at the Large Hadron Collider (LHC). The results are based on an integrated luminosity of approximately 7 μb(-1). Elliptic flow is measured over a wide region in pseudorapidity, vertical bar eta vertical bar < 2.5, and over a broad range in transverse momentum, 0.5 < p(T) < 20 GeV. The elliptic flow parameter nu(2) is obtained by correlating individual tracks with the event plane measured using energy deposited in the forward calorimeters. As a function of transverse momentum, nu(2)(p(T)) reaches a maximum at p(T) of about 3 GeV. then decreases and becomes weakly dependent on p(T) above 7-8 GeV. Over the measured pseudorapidity region, nu(2) is found to be only weakly dependent on eta, with less variation than observed at lower beam energies. The results are discussed in the context of previous measurements at lower collision energies, as well as recent results from the LHC.