  When binary black holes are embedded in a gaseous environment, a rotating
disk surrounding them, the so-called circumbinary disk, will be formed. The
binary exerts a gravitational torque on the circumbinary disk and thereby the
orbital angular momentum is transferred to it, while the angular momentum of
the circumbinary disk is transferred to the binary through the mass accretion.
The binary undergoes an orbital decay due to both the gravitational wave
emission and the binary-disk interaction. This causes the phase evolution of
the gravitational wave signal. The precise measurement of the gravitational
wave phase thus may provide information regarding the circumbinary disk. In
this paper, we assess the detectability of the signature of the binary-disk
interaction using the future space-borne gravitational wave detectors such as
DECIGO and BBO by the standard matched filtering analysis. We find that the
effect of the circumbinary disk around binary black holes in the mass range
$6M_sun\le{M}\lesssim3\times10^3M_sun$ is detectable at a statistically
significant level in five year observation, provided that gas accretes onto the
binary at a rate greater than $\dot{M}\sim1.4\times10^{17} [gs^{-1}]
j^{-1}(M/10M_sun)^{33/23}$ with 10% mass-to-energy conversion efficiency, where
j represents the efficiency of the angular momentum transfer from the binary to
the circumbinary disk. We show that $O(0.1)$ coalescence events are expected to
occur in sufficiently dense molecular clouds in five year observation. We also
point out that the circumbinary disk is detectable, even if its mass at around
the inner edge is by over 10 orders of magnitude less than the binary mass.
