We present a measurement of sigma B(W -> e nu) and sigma B(Z^0 ->
e^+e^-) in proton-antiproton collisions at sqrt s = 1.8 TeV using a
significantly improved understanding of the integrated luminosity.  The data
represent an integrated luminosity of 19.7 pb^-1 from the 1992-1993 run with
the Collider Detector at Fermilab.  We find sigma B(W -> e nu) = 2.49 +/- 0.12
nb and sigma B(Z^0 -> e^+e^-) = 0.231 +/- 0.012 nb.