  We present a survey of the Perseus molecular cloud in the J $=$
1$\rightarrow$0 transition of HCN, a widely used tracer of dense molecular gas.
The survey was conducted with the CfA 1.2 m telescope, which at 89 GHz has a
beam width of 11' and a spectral resolution of 0.85 km s$^{-1}$. A total of 8.1
deg$^2$ was surveyed on a uniform 10' grid to a sensitivity of 14 mK per
channel. The survey was compared with similar surveys of CO and dust in order
to study and calibrate the HCN line as a dense gas tracer. We find the HCN
emission to extend over a considerable fraction of the cloud. We show that the
HCN intensity remains linear with H$_2$ column density well into the regime
where the CO line saturates. We use radiative transfer modeling to show that
this likely results from subthermal excitation of HCN in a cloud where the
column and volume densities of H$_2$ are positively correlated. To match our
HCN observations the model requires an exponential decrease in HCN abundance
with increasing extinction, consistent with HCN depletion onto grains. The
modeling also reveals that the mean volume density of H$_2$ in the HCN emitting
regions is $\sim$ 10$^4$ cm$^{-3}$, well below the HCN critical density. For
the first time, we obtain a direct measurement of the ratio of dense gas mass
to HCN luminosity for an entire nearby molecular cloud: $\alpha$(HCN) $=$ 92
M$_\odot$/(K km s$^{-1}$ pc$^2$).
