  Using 448 million $\psi(2S)$ events, the spin-singlet $P$-wave charmonium
state $h_c$(1$^1P_1$) is studied via the $\psi(2S) \rightarrow \pi^0 h_c$ decay
followed by the $h_c \to \gamma \eta_c$ transition. The branching fractions are
measured to be $\mathcal{B}_{\rm Inc}(\psi(2S) \rightarrow \pi^0h_c) \times
\mathcal{B}_{\rm Tag}(h_c \rightarrow \gamma \eta_c) = (4.22 ^{+0.27}_{-0.26}
\pm 0.19) \times 10^{-4}$ , $\mathcal{B}_{\rm Inc}(\psi(2S) \rightarrow
\pi^0h_c) = (7.32 \pm 0.34 \pm 0.41) \times 10^{-4}$, and $\mathcal{B}_{\rm
Tag}(h_c \rightarrow \gamma \eta_c) = (57.66 ^{+3.62}_{-3.50} \pm 0.58)\%$,
where the uncertainties are statistical and systematic, respectively. The
$h_c$(1$^1P_1$) mass and width are determined to be $M = (3525.32 \pm 0.06 \pm
0.15)~\rm{MeV}/c^{\rm{2}}$ and ${\mathit \Gamma} = (0.78 ^{+0.27}_{-0.24} \pm
0.12)~\rm{MeV}$. Using the center of gravity mass of the three $\chi_{cJ}$(1$^3
P_J$) mesons ($M(\rm c.o.g.)$), the $1P$ hyperfine mass splitting is estimated
to be $\Delta_{\rm hyp} = M(h_c) - M(\rm c.o.g.) = (0.03 \pm 0.06 \pm 0.15) ~
\rm{MeV}/c^{\rm{2}}$, which is consistent with the expectation that the $1P$
hyperfine splitting is zero at the lowest-order.
