Abstract. Projective Steiner triple systems have the largest possible number of automorphisms. We study the question of how many Pasch trades suffice to convert a Projective Steiner triple system to a rigid (i.e. having only the trivial automorphism) one. In the paper we determine all 120 nonisomorphic systems obtained from the projective Steiner triple system of order 31 by at most three Pasch trades. Exactly three of these, each corresponding to three Pasch switches, are rigid. Thus three Pasch switches suffice, and are required, in order to convert the projective system of order 31 to a rigid one. This contrasts with the projective system of order 15 where four Pasch trades are required. We also show that four Pasch trades are required in order to turn the projective system of order 63 to a rigid system.