This paper presents the first measurement of the
differential $B^\pm$ and $B^0$ cross-sections, $d\sigma/dp_T$, in
$p\overline{p}$ collisions at $\sqrt{s}=1.8$ TeV.  The data sample used 
represents an integrated luminosity of $19.3 \pm 0.8$ $pb^{-1}$ accumulated by 
the Collider Detector at Fermilab(CDF).   
The cross-sections are measured over the $p_T$ range 6-15 GeV/c in the
central pseudorapidity region $|\eta| < 1$ by fully
reconstructing the $B$ meson decays $B^{\pm}\rightarrow J/\psi K^{\pm}$ and
$B^{0}\rightarrow J/\psi K^{*0}$, where the $J/\psi$ is required 
to decay to two muons, and the $K^{*0}$ is required to decay to 
$K^\pm \pi^\mp$.   The results are compared to the theoretical QCD 
prediction calculated at next-to-leading order.