  We present the results of an ALMA survey to identify 183 GHz H$_2$O maser
emission from AGN already known to host 22 GHz megamaser systems. Out of 20
sources observed, we detect significant 183 GHz maser emission from 13; this
survey thus increases the number of AGN known to host (sub)millimeter
megamasers by a factor of 5. We find that the 183 GHz emission is
systematically fainter than the 22 GHz emission from the same targets, with
typical flux densities being roughly an order of magnitude lower at 183 GHz
than at 22 GHz. However, the isotropic luminosities of the detected 183 GHz
sources are comparable to their 22 GHz values. For two of our sources -- ESO
269-G012 and the Circinus galaxy -- we detect rich 183 GHz spectral structure
containing multiple line complexes. The 183 GHz spectrum of ESO 269-G012
exhibits the triple-peaked structure characteristic of an edge-on AGN disk
system. The Circinus galaxy contains the strongest 183 GHz emission detected in
our sample, peaking at a flux density of nearly 5 Jy. The high signal-to-noise
ratios achieved by these strong lines enable a coarse mapping of the 183 GHz
maser system, in which the masers appear to be distributed similarly to those
seen in VLBI maps of the 22 GHz system in the same galaxy and may be tracing
the circumnuclear accretion disk at larger orbital radii than are occupied by
the 22 GHz masers. This newly identified population of AGN disk megamasers
presents a motivation for developing VLBI capabilities at 183 GHz.
