  Water-worlds are water-rich (>1 wt% H$_2$O) exoplanets. The classical models
of water-worlds considered layered structures determined by the phase
boundaries of pure water. However, water-worlds are likely to possess
comet-like compositions, with between ~3 mol% to 30 mol% CO$_2$ relative to
water. In this study, we build an interior structure model of habitable (i.e.
surface-liquid-ocean-bearing) water-worlds using the latest results from
experimental data on the CO$_2$-H$_2$O system, to explore the CO$_2$ budget and
to localize the main CO$_2$ reservoirs inside of these planets. We show that
CO$_2$ dissolved in the ocean and trapped inside of a clathrate layer cannot
accommodate a cometary amount of CO$_2$ if the planet accretes more than 11 wt%
of volatiles (CO$_2$ + H$_2$O) during its formation. We propose a new,
potentially dominant, CO$_2$ reservoir for water-worlds: CO$_2$ buried inside
of the high-pressure water ice mantle as CO$_2$ ices or (H$_2$CO$_3$ . H$_2$O),
monohydrate of carbonic acid. If insufficient amounts of CO$_2$ are sequestered
either in this reservoir or the planet's iron core, habitable zone water-worlds
could generically be stalled in their cooling before liquid oceans have a
chance to condense.
