We present a measurement of $\sigma \cdot B(W \rightarrow e \nu)$ and $\sigma \cdot B(Z~0 \rightarrow e~+e~-)$ 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 (CDF). We find $\sigma \cdot B(W \rightarrow e \nu) = 2.49 \pm 0.12$nb and $\sigma \cdot B(Z~0 \rightarrow e~+e~-) = 0.231 \pm 0.012$nb.
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