  Magnetic excitations in the spin-ladder material (C$_5$H$_{12}$N)$_2$CuBr$_4$
[BPCB] are probed by high-resolution multi-frequency electron spin resonance
(ESR) spectroscopy. Our experiments provide a direct evidence for a biaxial
anisotropy ($\sim 5\%$ of the dominant exchange interaction), that is in
contrast to a fully isotropic spin-ladder model employed for this system
previously. It is argued that this anisotropy in BPCB is caused by spin-orbit
coupling, which appears to be important for describing magnetic properties of
this compound. The zero-field zone-center gap in the excitation spectrum of
BPCB, $\Delta_0/k_{B}=16.5$ K, is detected directly. Furthermore, an ESR
signature of the inter-ladder exchange interactions is obtained. The detailed
characterization of the anisotropy in BPCB completes the determination of the
full spin hamiltonian of this exceptional spin-ladder material and shows ways
to study anisotropy effects in spin ladders.
