  The population of planets smaller than approximately $1.7~R_\oplus$ is widely
interpreted as consisting of rocky worlds, generally referred to as
super-Earths. This picture is largely corroborated by radial-velocity (RV) mass
measurements for close-in super-Earths but lacks constraints at lower
insolations. Here we present the results of a detailed study of the Kepler-138
system using 13 Hubble and Spitzer transit observations of the warm-temperate
$1.51\pm0.04~R_\oplus$ planet Kepler-138 d ($T_{\mathrm{eq, A_B=0.3}}$~350 K)
combined with new Keck/HIRES RV measurements of its host star. We find evidence
for a volatile-rich "water world" nature of Kepler-138 d, with a large fraction
of its mass contained in a thick volatile layer. This finding is independently
supported by transit timing variations, RV observations
($M_d=2.1_{-0.7}^{+0.6}~M_\oplus$), as well as the flat optical/IR transmission
spectrum. Quantitatively, we infer a composition of $11_{-4}^{+3}$\% volatiles
by mass or ~51% by volume, with a 2000 km deep water mantle and atmosphere on
top of a core with an Earth-like silicates/iron ratio. Any hypothetical
hydrogen layer consistent with the observations ($<0.003~M_\oplus$) would have
swiftly been lost on a ~10 Myr timescale. The bulk composition of Kepler-138 d
therefore resembles those of the icy moons rather than the terrestrial planets
in the solar system. We conclude that not all super-Earth-sized planets are
rocky worlds, but that volatile-rich water worlds exist in an overlapping size
regime, especially at lower insolations. Finally, our photodynamical analysis
also reveals that Kepler-138 c ($R_c=1.51 \pm 0.04~R_\oplus$,
$M_c=2.3_{-0.5}^{+0.6}~M_\oplus$) is a slightly warmer twin of Kepler-138 d,
i.e., another water world in the same system, and we infer the presence of
Kepler-138 e, a likely non-transiting planet at the inner edge of the habitable
zone.
