We report on a search for pair production of first-generation scalar leptoquarks ($LQ$) in $p bar{p}$ collisions at $sqrt{s}$=1.96 TeV using an integrated luminosity of 203 $pb^{-1}$ collected at the Fermilab Tevatron collider by the CDF experiment. We observe no evidence for $LQ$ production in the topologies arising from $LQ bar{LQ} to eqeq$ and $LQ bar{LQ} to eq u q$, and derive 95% C.L. upper limits on the $LQ$ production cross section. %as a function of $beta$, where $beta$ is the branching fraction for $LQ to eq$. The results are combined with those obtained from a separately reported CDF search in the topology arising from $LQbar{LQ} to u q u q$ and 95% C.L. lower limits on the LQ mass as a function of $beta = BR(LQ to eq) $ are derived. The limits are 236, 205 and 145 GeV/c$^2$ for $beta$ = 1, $beta$ = 0.5 and $beta$ = 0.1, respectively.