  We investigate the decay B^+ -> J/psi phi K^+ in a search for the X(4140)
state, a narrow threshold resonance in the J/psi phi system. The data sample
corresponds to an integrated luminosity of 10.4 fb^{-1} of ppbar collisions at
sqrt s =1.96 TeV collected by the D0 experiment at the Fermilab Tevatron
collider. We observe a mass peak with a statistical significance of 3.1
standard deviations and measure its invariant mass to be M=4159.0 +-.3 (stat)
+- 6.6 (syst) MeV and its width to be Gamma= 19.9+-12.6 (stat)}^{+1.0}_{-8.0}
(syst) MeV. Identifying this mass peak as the X(4140), we measure, for M(J/psi
phi) <4.59 GeV, the relative branching fraction B_rel= B(B^+ -> X(4140) K^+)/
B(B^+ ->J/psi phi K^+) to be 21+-8 (stat) +- 4 (syst)%. In addition, the data
can accommodate the presence of a second enhancement at a mass of 4328.5+- 12.0
MeV.
