Using a T=0 real-space renormalisation-group method, the spontaneous magnetisation is calculated on an impurity site of the quantum spin-1/2 Ising chain in a transverse field. Two kinds of impurities are considered: either a transverse field or a bond is different from its host value. In both cases the magnetisation vanishes at the critical field with an exponent beta 0 different from the host exponent and varying continuously with the impurity field (or impurity bond). These results agree with recent exact investigations of equivalent classical two-dimensional models.