  We present a measurement of the abundance of carbon monoxide in the early
Universe, utilizing the final results from the CO Power Spectrum Survey
(COPSS). Between 2013 and 2015, we performed observations with the
Sunyaev-Zel'dovich Array (SZA) to measure the aggregate CO emission from
$z\sim3$ galaxies with the intensity mapping technique. Data were collected on
19 fields, covering an area of 0.7 square degrees, over a frequency range of 27
to 35 GHz. With these data, along with data analyzed in COPSS I, we are able to
observe the CO(1-0) transition in the redshift range of $z=2.3-3.3$ for spatial
frequencies between $k=0.5-10\ h\,\textrm{Mpc}^{-1}$, spanning a comoving
volume of $4.9\times10^{6}\ h^{-3}\,\textrm{Mpc}^{3}$. We present estimates of
contributions from continuum sources and ground illumination within our
measurement. We constrain the amplitude of the CO power spectrum to be
$P_{\textrm{CO}}=3.0_{-1.3}^{+1.3}\times10^{3}\ \mu\textrm{K}^{2}
(h^{-1}\,\textrm{Mpc})^{3}$, or $\Delta^{2}_{\textrm{CO}}(k\!=\!1\
h\,\textrm{Mpc}^{-1})=1.5^{+0.7}_{-0.7} \times10^{3}\ \mu\textrm{K}^{2}$, at
68% confidence, and $P_{\textrm{CO}}>0$ at 98.9% confidence. These results are
a factor of 10 improvement in sensitivity compared to the previous limits set
in COPSS I. We use this measurement to place constraints on the CO(1-0) galaxy
luminosity function at $z\sim3$. We constrain the ratio of CO(1-0) luminosity
to host halo mass to $A_{\textrm{CO}}=6.3_{-2.1}^{+1.4}\times10^{-7}\
L_{\odot}\ M_{\odot}^{-1}$, and estimate a mass fraction of molecular gas of
$f_{\textrm{H}_{2}}=5.5^{+3.4}_{-2.2}\times10^{-2}$ for halos with masses of
order $10^{12}M_{\odot}$. Using simple theoretical estimates for the scaling of
molecular gas mass fraction and halo mass, we estimate the global density of
molecular gas to be
$\rho_{z\sim3}(\textrm{H}_{2})=1.1_{-0.4}^{+0.7}\times10^{8}\ M_{\odot}\
\textrm{Mpc}^{-3}$.
