  The first observation of the B0->J/\psi K+K- decay is presented with a data
sample corresponding to an integrated luminosity of 1.0/fb of pp collisions at
a center-of-mass energy of 7 TeV collected with the LHCb detector. The
branching fraction is measured to be B(B0->J/\psi K+K-) = (2.53 +/- 0.31 +/-
0.19)x10^{-6}, where the first uncertainty is statistical and the second is
systematic. An amplitude analysis of the final state in the B0->J/\psi K+K-
decay is performed to separate resonant and nonresonant contributions in the
K+K- spectrum. Evidence of the a0(980) resonance is reported with statistical
significance of 3.9 standard deviations. The corresponding product branching
fraction is measured to be B(B0->J/\psi a0(980), a0(980)->K+K-)=(4.70 +/- 3.31
+/- 0.72)x10^{-7}, yielding an upper limit of B(B0->J/\psi a0(980),
a0(980)->K+K-)<9.0x10^{-7} at 90% confidence level. No evidence of the resonant
decay B0->J/\psi\phi\; is found, and an upper limit on its branching fraction
is set to be B(B0->J/\psi\phi)< 1.9x10^{-7} at 90% confidence level.
