The flag geometry 1=(P, L, I) of a finite projective plane 6 of order s is the generalized hexagon of order (s, 1) obtained from 6 by putting P equal to the set of all flags of 6, by putting L equal to the set of all points and lines of 6, and where I is the natural incidence relation (inverse containment), i.e., 1 is the dual of the double of 6 in the sense of H. Van Maldeghem (1998, ``Generalized Polygons,'' Birkha user Verlag, Basel). Then we say that 1 is fully and weakly embedded in the finite projective space PG(d, q) if 1 is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of 1 generates PG(d, q), and if the set of points of 1 not opposite any given point of 1 does not generate PG(d, q). In an earlier paper, we have shown that the dimension d of the projective space belongs to [6, 7, 8], and that the projective plane 6 is Desarguesian. Furthermore, we have given examples for d=6, 7. In the present paper we show that for d=6, only these examples exist, and we also partly handle the case d=7. More precisely, we completely classify the full and weak embeddings of 1 (1 as above) in the case that there are two opposite lines L, M of 1 with the property that the subspace of PG(d, q) generated by all lines of 1 meeting either L or M has dimension 6 (which is the case for all embeddings in PG(d, q), d # [6, 7]). Together with Parts 2 and 3, this will provide the complete classification of all full and weak embeddings of 1.