  Detecting neutral Hydrogen (H\,{\sc i}) via the 21~cm line emission from the
intergalactic medium at $z\gtrsim 6$ has been identified as one of the most
promising probes of the epoch of cosmic reionization -- a major phase
transition of the Universe. However, these studies face severe challenges
imposed by the bright foreground emission from cosmic objects. Current
techniques require precise instrumental calibration to separate the weak
H\,{\sc i} line signal from the foreground continuum emission. We propose to
mitigate this calibration requirement by using measurements of the
interferometric bi-spectrum phase. Bi-spectrum phase is unaffected by
antenna-based direction-independent calibration errors and hence for a compact
array it depends on the sky brightness distribution only (subject to the usual
thermal-like noise). We show that the bi-spectrum phase of foreground
synchrotron continuum has a characteristically smooth spectrum relative to the
cosmological line signal. The two can be separated effectively by exploiting
this spectral difference using Fourier techniques, while eliminating the need
for precise antenna-based calibration of phases introduced by the instrument,
and the ionosphere, inherent in existing approaches. Using fiducial models for
continuum foregrounds, and for the cosmological H\,{\sc i} signal, we show the
latter should be detectable in bi-spectrum phase spectra, with reasonable
significance at $|k_\parallel| \gtrsim 0.5\,h$~Mpc$^{-1}$, using existing
instruments. Our approach will also benefit other H\,{\sc i} intensity mapping
experiments that face similar challenges, such as those measuring Baryon
Acoustic Oscillations (BAO).
