  We report the results of a search for the rare baryonic decays $B^0 \to
p\bar{p}$, $\Lambda\bar{\Lambda}$, and $B^+ \to p\bar{\Lambda}$. The analysis
is based on a data set of $31.7\times 10^6 B\bar{B}$ events collected by the
Belle detector at the KEKB $e^+e^-$ collider. No statistically significant
signals are found, and we set branching fraction upper limits ${\mathcal B}(B^0
\to p\bar{p}) < 1.2 \times 10^{-6}$, ${\mathcal B}(B^0 \to
\Lambda\bar{\Lambda}) < 1.0 \times 10^{-6}$, and ${\mathcal B}(B^+ \to
p\bar{\Lambda}) < 2.2 \times 10^{-6}$ at the 90% confidence level.
