A measurement of W --> e nu + n-jet cross sections in p anti-p collisions at s**(1/2) = 1.96 TeV using the Collider Detector at Fermilab in Run II is presented. The measurement is based on integrated luminosity of 320 pb-1, and includes events with jet multiplicity from > 1 to > 4. In each jet multiplicity sample the differential and cumulative cross sections with respect to the transverse energy of the nth-leading jet are measured. For W + > 2 jets the differential cross section with respect to the 2-leading jets invariant mass m_{j_1 j_2} and angular separation Delta R_{j_1 j_2} is also reported. The data are compared to predictions from Monte Carlo simulation.