  Based on a sample of $(1310.6 \pm 7.0) \times 10^6~J/\psi$ events collected
with the BESIII detector, we present measurements of $J/\psi$ and $\eta^\prime$
absolute branching fractions using the process
$J/\psi\rightarrow\gamma\eta^\prime$. By analyzing events where the radiative
photon converts into an $e^+e^-$ pair, the branching fraction for $J/\psi
\rightarrow \gamma \eta^\prime$ is measured to be $(5.27\pm0.03\pm0.05)\times
10^{-3}$. The absolute branching fractions of the five dominant decay channels
of the $\eta^\prime$ are then measured independently for the first time and are
determined to be $\mathcal{B}(\eta^\prime \rightarrow \gamma \pi^{+} \pi^{-})$
= (29.90$\pm$0.03$\pm$0.55)%, $\mathcal{B}(\eta^\prime \rightarrow \eta \pi^{+}
\pi^{-})$ = (41.24$\pm$0.08$\pm$1.24)%, $\mathcal{B}(\eta^\prime \rightarrow
\eta \pi^{0} \pi^{0})$ = (21.36$\pm$0.10$\pm$0.92)%, $\mathcal{B}(\eta^\prime
\rightarrow \gamma \omega)$ = (2.489$\pm$0.018$\pm$0.074)%, and
$\mathcal{B}(\eta^\prime \rightarrow \gamma \gamma)$ =
(2.331$\pm$0.012$\pm$0.035)%, where the first uncertainties are statistical and
the second systematic.
