As far as I know Dukat never traveled to the gamma quadrant through the wormhole, so how did he reach them to present his offer for Cardassia to join the Dominion?
Although the two parties are never shown making their deal on-screen, the fourth and final story in The Badlands duology does discuss the arrangement with some minor detail.
Apparently, Dukat had initially contacted the Dominion via subspace, thus initiating the talks. After several communications via subspace, Weyoun 5 secretly met with Dukat in person to finalize the agreement. The finalization occurred during the time between episode S5E13 "For the Uniform" and S5E14 "In Purgatory's Shadow". The signing of the alliance was successful, but ended on a sour note when the USS Defiant pursued a Romulan vessel into the section of The Badlands where Dukat and Weyoun's ships were hidden.
From dialogue in later episodes, it's apparent that Dukat promised not only Cardassia as an ally, but also the re-capture of Deep Space Nine - a major strategic goal of The Dominion. Conversely, The Dominion promised that Dukat would be installed as head of the new Cardassian government and that they would help Cardassia regain its lost territory (including Bajor). All of these events/promises were discussed & played out on-screen over the next three seasons.