  We explore the possibilities for a next-generation
electron-electric-dipole-moment experiment using ultracold heteronuclear
diatomic molecules assembled from a combination of radium and another
laser-coolable atom. In particular, we calculate their ground state structure
and their sensitivity to parity- and time-reversal (${\cal{P,T}}$) violating
physics arising from flavor-diagonal charge-parity (${\cal{CP}}$) violation.
Among these species, the largest ${\cal{P,T}}$-violating molecular interaction
constants -- associated for example with the electron electric dipole moment --
are obtained for the combination of radium (Ra) and silver (Ag) atoms. A
mechanism for explaining this finding is proposed. We go on to discuss the
prospects for an electron EDM search using ultracold, assembled, optically
trapped RaAg molecules, and argue that this system is particularly promising
for rapid future progress in the search for new sources of ${\cal{CP}}$
violation.
