We report the first measurement of charged particle elliptic flow in Pb-Pb collisions at sqrt[S(NN)] =2.76 TeV with the ALICE detector at the CERN Large Hadron Collider. The measurement is performed in the central pseudorapidity region (|η|<0.8) and transverse momentum range 0.2<p t<5.0 GeV/c. The elliptic flow signal v₂, measured using the 4-particle correlation method, averaged over transverse momentum and pseudorapidity is 0.087 ± 0.002(stat) ± 0.003(syst) in the 40%-50% centrality class. The differential elliptic flow v₂ p t reaches a maximum of 0.2 near p t =3 GeV/c. Compared to RHIC Au-Au collisions at sqrt[S(NN)] 200 GeV, the elliptic flow increases by about 30%. Some hydrodynamic model predictions which include viscous corrections are in agreement with the observed increase.