  A linear correlation has been proposed between the CO luminosity
($\rm{L}^{\prime}_{\rm{CO}}$) and full-width at half maximum (FWHM) for
high-redshift (z > 1) submillimeter galaxies. However, the controversy
concerning the $\rm{L}^{\prime}_{\rm{CO}}$-FWHM correlation seems to have been
caused by the use of heterogeneous samples (e.g., different transition lines)
and/or data with large measurement uncertainties. In order to avoid the
uncertainty caused by using different rotational transitions, in this work we
make an extensive effort to select only CO($J = 1-0$) data from the literature.
We separate these wide-ranging redshift data into two samples : the
low-redshift (z < 1) and high-redshift (z > 1) samples. The samples are
corrected for lensing magnification factors if gravitational-lensing effects
appeared in the observations. The correlation analysis shows that there exists
significant $\rm{L}^{\prime}_{\rm{CO}}$-FWHM correlations for both the
low-redshift and high-redshift samples. A comparison of the low- and
high-redshift $\rm{L}^{\prime}_{\rm{CO}}$-FWHM correlations does not show
strong evolution with redshift. Assuming that there is no evolution, we can use
this relation to determine the model independent distances of high-redshift
galaxies. We then constrain cosmological models with the calibrated
high-redshift CO data and the sample of Type Ia supernovae in the Union 2.1
compilation. In the constraint for wCDM with our samples, the derived values
are w_{0} = -1.02 {\pm} 0.17, {\Omega}_{m0} = 0.30{\pm}0.02, and H_{0} = 70.00
{\pm}0.60 km\,s^{-1}\,Mpc^{-1}.
