  We show that the $\kappa$-deformed Poincar\'e quantum algebra proposed for
elementary particle physics has the structure of a Hopf agebra bicrossproduct
$U(so(1,3))\cobicross T$. The algebra is a semidirect product of the classical
Lorentz group $so(1,3)$ acting in a deformed way on the momentum sector $T$.
The novel feature is that the coalgebra is also semidirect, with a backreaction
of the momentum sector on the Lorentz rotations. Using this, we show that the
$\kappa$-Poincar\'e acts covariantly on a $\kappa$-Minkowski space, which we
introduce. It turns out necessarily to be deformed and non-commutative. We also
connect this algebra with a previous approach to Planck scale physics.
