Bashir plays reluctant host to a delegation of Federation ambassadors, one of whom happens to be Lwaxana Troi, mother of the Enterprise's counselor, Deanna Troi. Lwaxana creates a scene in Quark's when she discovers her priceless hairbrooch has been stolen, but Odo catches the culprit and returns the heirloom to her. The Betazoid develops an instant attraction to the shape-shifter. Meanwhile, much to Sisko's chagrin, Bashir brings the other three ambassadors to Ops, where a frustrated O'Brien struggles with the station's uncooperative computer. Then, an unidentified vessel comes through the wormhole. Unoccupied, it appears to be an alien probe. Sisko has it towed to a position a few hundred meters from the docking ring, while they attempt to establish a link to download information. Elsewhere, Lwaxana, dressed to kill, makes a move on Odo, who upon realizing her intentions, nervously avoids her and makes a beeline for Ops.
Much to O'Brien's surprise, getting the station's computer to perform a data transfer from the probe is unusually easy. While he and Dax work with the data, Odo seeks Sisko's advice about Lwaxana's romantic advances, but finds Sisko amused that the shape-shifter can handle thieves and killers but not an amorous Betazoid woman. Odo returns to the Promenade, runs right into Lwaxana, then tries to escape into the turbolift, but she gets in with him. Suddenly, it stops between floors — the power has failed. Odo signals to be beamed out, but then the transporter also mysteriously goes dead. Lwaxana finally has Odo where she wants him — alone at last.
O'Brien is baffled to find that both the turbolift and the transporter are in working order, even though they aren't working. Told that he can't even shape-shift his way out, Odo is resigned to being stuck with Lwaxana for a while. Odo suggests that he and the Betazoid pass the time quietly, but no such luck — Lwaxana rambles on and on. Back in Ops, O'Brien informs Sisko that the computer rerouted the E-P-S power flow uncharacteristically fast, but for some reason still didn't activate the turbolift circuits. O'Brien's gut instincts tell him that, strange as the notion sounds, the computer's personality has changed — becoming more obedient, and performing as if it wants constant attention — much like ... a child.
Sisko, Kira, O'Brien, and Dax discuss the mystery, and a theory develops that they may have actually downloaded a non-biological lifeform into the computer. O'Brien suggests that transferring all of that downloaded programming off the station might purge the lifeform, and sets to attempt the procedure, but the computer seems to be stopping him at every turn. In the turbolift, meanwhile, Odo shares some private thoughts about his background with a sympathetic Lwaxana, who notices that he isn't looking well. He reminds her that he must revert to a liquid state every sixteen hours — and this is now hour fifteen. While time continues to tick away, O'Brien attempts a manual transfer out of the computer, but sparks suddenly fly from his console during the process, followed by a plasma explosion around the station's guest quarters. Fire breaks out in the corridor, trapping Bashir and the three ambassadors.
While Sisko, Kira, and a rescue team try to cut their way into the corridor, O'Brien acts on a hunch and begins to construct a subprogram within the computer that will attract the entity, which he feels is actually acting like a stray puppy. In the turbolift, Odo cannot hold his shape any longer, so Lwaxana lifts her dress to form an improvised basin, into which Odo dissolves safely to his liquid form. Back in Ops, O'Brien has the computer route all back-up functions through his new subprogram, after which all systems come back on line. Sisko and Kira, now able to get into the fire-scorched corridor, search through the rubble, and are surprised when Bashir and the ambassadors come out from a wall compartment, soiled but safe. Lwaxana and Odo, now back in his humanoid form, are freed from the turbolift, and she leaves him with a hint of romance to follow when they meet next, O'Brien explains to Sisko that the subprogram he built will keep their "adopted pet" lifeform happy, busy, and out of their way from now on.