We present constraints on the abundance of carbon-monoxide in the early Universe from the CO Power Spectrum Survey (COPSS). We utilize a data set collected between 2005 and 2008 using the Sunyaev-Zeldovich Array (SZA), which were previously used to measure arcminute-scale fluctuations of the CMB. This data set features observations of 44 fields, covering an effective area of 1.7 square degrees, over a frequency range of 27 to 35 GHz. Using the technique of intensity mapping, we are able to probe the CO(1-0) transition, with sensitivity to spatial modes between $k=0.5{-}2 h,textrm{Mpc}^{-1}$ over a range in redshift of $z=2.3{-}3.3$, spanning a comoving volume of $3.6times10^{6} h^{-3},textrm{Mpc}^{3}$. We demonstrate our ability to mitigate foregrounds, and present estimates of the impact of continuum sources on our measurement. We constrain the CO power spectrum to $P_{textrm{CO}}<2.6times10^{4} mutextrm{K}^{2} (h^{-1},textrm{Mpc})^{3}$, or $Delta^{2}_{textrm{CO}}(k! = ! 1 h,textrm{Mpc}^{-1})<1.3 times10^{3} mutextrm{K}^{2}$, at $95%$ confidence. This limit resides near optimistic predictions for the CO power spectrum. Under the assumption that CO emission is proportional to halo mass during bursts of active star formation, this corresponds to a limit on the ratio of $textrm{CO}(1{-}0)$ luminosity to host halo mass of $A_{textrm{CO}}<1.2times10^{-5} L_{odot} M_{odot}^{-1}$. Further assuming a Milky Way-like conversion factor between CO luminosity and molecular gas mass ($alpha_{textrm{CO}}=4.3 M_{odot} (textrm{K} textrm{km} textrm{s}^{-1} textrm{pc}^{-2})^{-1}$), we constrain the global density of molecular gas to $rho_{zsim3}(M_{textrm{H}_{2}})leq 2.8 times10^{8} M_{odot} textrm{Mpc}^{-3}$.
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