A search for the rare decays of B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] and B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] is presented in p p collis d √ sions at s=1.96 TeV corresponding to 10 fb[-]1 of integrated luminosity collected by the CDF II detector. The observed number of B0 candidates is consistent with d background-only expectations and yields an upper limit on the branching fraction B( B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] ) of 4.2 x 10[-]9 at 95% confidence level. An excess is observed d for the B0 signal region. The probability that background processes alone could s produce such an excess or larger is 0.94%. The probability that the combination of background and the expected standard model rate B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] could produce s such an excess or larger is 6.8%. These data are used to determine a central value of B( B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] ) = (1.3+0..9 ) x 10[-]8 as well as a two sided limit at 90% C.L. s [-]0 7 of 2.2 x 10[-]9 < B( B0 [RIGHTWARDS ARROW] [MICRO SIGN]+ [MICRO SIGN][-] ) < 3.0 x 10[-]8 . s