  Data suggest that most rocky exoplanets with orbital period $p$ $<$ 100 d
("hot" rocky exoplanets) formed as gas-rich sub-Neptunes that subsequently lost
most of their envelopes, but whether these rocky exoplanets still have
atmospheres is unknown. We identify a pathway by which 1-1.7 $R_{Earth}$ (1-10
$M_{Earth}$) rocky exoplanets with orbital periods of 10-100 days can acquire
long-lived 10-2000 bar atmospheres that are H$_2$O-dominated, with mean
molecular weight $>$10. These atmospheres form during the planets' evolution
from sub-Neptunes into rocky exoplanets. H$_2$O that is made by reduction of
iron oxides in the silicate magma is highly soluble in the magma, forming a
dissolved reservoir that is protected from loss so long as the H$_2$-dominated
atmosphere persists. The large size of the dissolved reservoir buffers the
H$_2$O atmosphere against loss after the H$_2$ has dispersed. Within our model,
a long-lived, water-dominated atmosphere is a common outcome for efficient
interaction between a nebula-derived atmosphere (peak atmosphere mass fraction
0.1-0.6 wt%) and oxidized magma ($>$5 wt% FeO), followed by atmospheric loss.
This idea predicts that most rocky planets that have orbital periods of 10-100
days and that have radii within 0.1-0.2 $R_{Earth}$ of the lower edge of the
radius valley still retain H$_2$O atmospheres. This prediction is imminently
testable with JWST and has implications for the interpretation of data for
transiting super-Earths.
