That's difficult one, Cattie.  Dismissal is a minefield, and you need to ensure that you don't do anything that can result in an employment tribunal hearing.

"Effectively her Line Manager" is an interesting phrase.  Her Line Manager should be stated somewhere in her contract, and that is the person who should be having this conversation with her.

First of all, have a look at your company grievance/disciplinary procedures and follow those.

Assuming you are this person's Line Manager ... here is a basic structure as to how I would handle such matters.

Step one
Arrange to meet Miss X at an appointed time.  Say that you wish to discuss work issues with her on an informal basis, but allow her the opportunity of trade union representation (if relevant to your company) or to be accompanied by a friend.

At the meeting, state that the status of the meeting is informal, with no further action intended at this stage.  You have been asked to speak with her about her performance.  List the ways in which she has not achieved the required performance, and point to evidence.  Ask her, after each point, for her comments. There may be all sorts of reasons why she is not performing well, including domestic matters.  Offer support, counselling, occupational health - whatever resources that are available and appropriate.  Make notes as you go along.

At the conclusion of the meeting, draw up TOGETHER, a plan for improvement and target dates for improvement.  

Type up the notes, give them to her to agree and sign and return to you (with a copy for herself) and file them in her file.

Step two
Allow her the opportunity of trade union representation (if relevant to your company) or to be accompanied by a friend.  Explain that the status of the meeting is formal. This is the verbal warning stage (you could do this at step one, but I assume that you would like this person to shape up and stay).

Step three
Written warning.

Step four
Dismissal

At all stages, copies of notes and letters, memos etc relating to this matter should be filed in the staff member's personnel file.  

Hope this is of some help.

Best wishes,

Raindance