A search for the Standard Model Higgs boson was carried out in WH 2!l[nu]b{bar b} process in p{bar p} collisions at a center of mass energy of 1.96 TeV, where W, H, l, [nu], b and p denote either a W or W− boson, Higgs boson, lepton (electron or muon), neutrino, bottom quark and proton, respectively. The data were collected with the Collider Detector at Fermilab from February 2002 to August 2004. The corresponding integrated luminosity is 319 pb−1. We select events containing a single high-p{sub T} electron or muon, a large imbalance of the total transverse energy from a neutrino and two b quark jets. The main backgrounds are the W + light flavor/gluon jets and W + heavy flavor jets processes. Requiring the secondary vertex b-tagging enables us to reject the W + light flavor/gluon jets events effectively. After all event selections, they observe 187 events which is in agreement with the Standard Model background expectation of 175.2 ± 26.3 events, and there is no significant excess originating from the Higgs boson in the reconstructed dijet invariant mass distribution. They thus set a 95% confidence level upper limit on the production cross section times branching ratio decaying into bb, [sigma](p{bar p} 2!W H) x Br(H 2!b{bar b}). The detection efficiency for the W H events, which is necessary for calculating the upper limit, is determined by the Monte Carlo except for the lepton identification efficiency, the lepton trigger efficiency and the b-tagging efficiency which are estimated from the CDF real data. The resultant 95% confidence level upper limits are 10.0 pb to 2.8 pb using at least one b-tagging method and 9.7 pb to 6.6 pb using double b-tagging method for the Higgs boson mass region 110 GeV/c2 to 150 GeV/c2, where the Standard Model prediction is approximately one or two order of magnitude lower than the results.