  We present the first measurements of the absolute branching fractions of
$\Xi_c^+$ decays into $\Xi^- \pi^+ \pi^+$ and $p K^- \pi^+$ final states. Our
analysis is based on a data set of $(772\pm 11)\times 10^{6}$ $B\bar{B}$ pairs
collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB
$e^+e^-$ collider. We measure the absolute branching fraction of $\bar{B}^{0}
\to \bar{\Lambda}_{c}^{-} \Xi_{c}^{+}$ with the $\Xi_c^+$ recoiling against
$\bar{\Lambda}_c^-$ in $\bar{B}^0$ decays resulting in ${\cal B}(\bar{B}^{0}
\to \bar{\Lambda}_{c}^{-} \Xi_{c}^{+}) = [1.16 \pm 0.42(\rm stat.) \pm 0.15(\rm
syst.)] \times 10^{-3}$. We then measure the product branching fractions ${\cal
B}(\bar{B}^{0} \to \bar{\Lambda}_c^- \Xi_c^+){\cal B}(\Xi_c^+ \to \Xi^- \pi^+
\pi^+)$ and ${\cal B}(\bar{B}^{0} \to \bar{\Lambda}_c^- \Xi_c^+){\cal
B}(\Xi_c^+ \to p K^- \pi^+)$. Dividing these product branching fractions by
$\bar{B}^{0} \to \bar{\Lambda}_{c}^{-} \Xi_{c}^{+}$ yields: ${\cal B}(\Xi_c^+
\to \Xi^- \pi^+ \pi^+) = [2.86 \pm 1.21(\rm stat.) \pm 0.38(\rm syst.)]\%$ and
${\cal B}(\Xi_c^+ \to p K^- \pi^+)=[0.45 \pm 0.21(\rm stat.) \pm 0.07(\rm
syst.)]\%$. Our result for ${\cal B}(\Xi_c^+ \to \Xi^- \pi^+ \pi^+)$ can be
combined with $\Xi_c^+$ branching fractions measured relative to $\Xi_c^+ \to
\Xi^- \pi^+ \pi^+$ to set the absolute scale for many $\Xi_c^+$ branching
fractions.
