  CsFeBr$_3$ is an $S\,{=}\,1$ hexagonal antiferromagnet that has a singlet
ground state owing to its large easy-plane single-ion anisotropy. The critical
behavior of the magnetic-field-induced phase transition for a magnetic field
parallel to the $c$ axis, which can be described by the Bose-Einstein
condensation (BEC) of magnons under the $U(1)$ symmetry, was investigated via
magnetization and specific heat measurements down to 0.1$\,$K. For the specific
heat measurement, we have developed a method of effectively suppressing the
torque acting on a sample with strong anisotropy that uses the spin dimer
compound Ba$_2$CoSi$_2$O$_6$Cl$_2$ with large and anisotropic Van Vleck
paramagnetism. The temperature dependence of the transition field $H_{\rm
c}(T)$ was found to follow the power-law $H_{\rm c}(T)\,{-}\,H_{\rm
c}\,{\propto}\,T^{\phi}$ with a critical exponent of
${\phi}\,{=}\,1.50\,{\pm}\,0.02$ and critical field of $H_{\rm c}\,{=}\,2.60$
T. This result verifies the universality of the three-dimensional BEC of
magnons described by ${\phi}_{\rm BEC}\,{=}\,3/2$.
