The invention concerns a method and an apparatus in a paper machine, in particular in its drying section, for intensification of the drying of the paper. In the method the paper web (W) is passed on support of a drying fabric onto a drying cylinder (K). In the method, in connection with the drying cylinder (K), a separate suction device (10) is employed, which is fitted at a distance (F) from the drying fabric, e.g. a wire (H), which follows the face of the drying cylinder. In the method, by means of the suction device (10), the boundary layer (S.sub.1) of moist air, which has been formed on the face of the drying fabric (H), is suctioned off, whereby the drying process is promoted.