A sulfur-crosslinkable adhesive composition of the non-aqueous type, that is to say not requiring the presence of water, usable for direct bonding of a pregelled polyester or vinyl ester composite to rubber in the uncured state. The composition comprises polyvinylpyridine-styrene-butadiene elastomer (“p-VSBR”) in combination with a polyester or vinyl ester resin. Preferably, the resin used is a novolak and/or bisphenol based resin of the epoxy vinyl ester type. This composition, unlike conventional RFL adhesives, does not require any activation treatment, such as intermediate drying, at the time of use. A process for preparing such an adhesive composition, use of said composition for the manufacture of laminated composite parts.