Edit 2:
Finally going through this part of the game myself (Romancing Ashley in 1) I had a different set of dialogue choices that were Paragon/Renegade. Going with the Paragon interrupt (have your squad stand down) and then choosing the Paragon dialogue option, I saved Ash's life. I can only assume if I did not do the interrupt and/or chose the Renegade option, she would not have survived
Ashley is unaware that Udina and Cerberus are working together to lead the Councilors into a trap so Udina can take control of the Council. This puts Shepard and Ashley in conflict; Shepard must then convince Ashley into realizing the truth about what's going on and turning on Udina instead. If Shepard cannot persuade Ashley to back off, then Shepard can kill Ashley. If not, a squad member will shoot her.
Edit:
In this walkthrough, it shows that it seems the initial dialogue choice and whether or not you choose the Paragon interrupt determines if Ashley/Kaiden lives or dies during this encounter. This walk through also did not Romance Ashley in ME1, but Liara, which explains the different dialogue options than mine above.