  We present ALMA CO (2-1) detections of 24 star-forming Brightest Cluster
Galaxies (BCGs) over $0.2<z<1.2$, constituting the largest and most distant
sample of molecular gas measurements in BCGs to date. The BCGs are selected
from the Spitzer Adaptation of the Red-Sequence Cluster Survey (SpARCS) to be
IR-bright and therefore star-forming. We find that molecular gas is common in
star-forming BCGs, detecting CO at a detection rate of 80% in our target sample
of 30 objects. We additionally provide measurements of the star formation rate
(SFR) and stellar mass, calculated from existing MIPS 24 $\mu$m and IRAC 3.6
$\mu$m fluxes, respectively. We find these galaxies have molecular gas masses
of $0.7-11.0\times 10^{10}\ \mathrm{M}_\odot$, comparable to other BCGs in this
redshift range, and specific star formation rates which trace the Elbaz et al.
(2011) Main Sequence. We compare our BCGs to those of the lower-redshift,
cooling-flow BCG sample assembled by Edge (2001) and find that at z $\lesssim
0.6$ the two samples show very similar correlations between their gas masses
and specific SFRs. We suggest that, in this redshift regime, the $\sim10\%$
(Webb et al., 2015) of BCGs that are star-forming process any accreted
molecular gas into stars through means that are agnostic to both their redshift
and their cluster mass.
