  The formation of planets in compact or highly eccentric binaries and the
migration of hot Jupiters are two outstanding problems in planet formation. The
hot Jupiter $\tau$ Boo Ab orbits the primary star in the long-period ($P
\gtrsim 1000\,$yr), highly eccentric ($e \sim 0.9$) binary $\tau$ Bo\"otis. Due
to the long orbital period, the orbit of the stellar binary is poorly
constrained. Here we aim to constrain the orbit of the stellar binary $\tau$
Boo AB in order to investigate the formation and migration history of the
system. The mutual orbital inclination of the stellar companion and the hot
Jupiter has important implications for planet migration. The binary
eccentricity and periastron distance are important for understanding the
conditions under which $\tau$ Boo Ab formed. We combine more than 150 years of
astrometric data with twenty-five years of high precision radial velocities.
The combination of sky-projected and line-of-sight measurements places tight
constraints on the orbital inclination, eccentricity, and periastron distance
of $\tau$ Boo AB. We find an orbital inclination of $47.2^{+2.7}_{-3.7}\,$deg,
periastron distance of $28.3^{+2.3}_{-3.0}\,$au and eccentricity of
$0.87^{+0.04}_{-0.03}$. We find that the orbital inclinations of $\tau$ Boo Ab
and $\tau$ Boo B, as well as the stellar spin-axis of $\tau$ Boo A coincide at
$\sim \! 45$ degrees, a result consistent with the assumption of a
well-aligned, coplanar system. The likely aligned, coplanar configuration
suggests planetary migration within a well-aligned protoplanetary disc. Due to
the high eccentricity and small periastron distance of $\tau$ Boo B, the
protoplanetary disc was tidally truncated at $\approx \! 6\,$au. We suggest
that $\tau$ Boo Ab formed near the edge of the truncated disc and migrated
inwards with high eccentricity due to spiral waves generated by the stellar
companion.
