Story is impromptu one with no depth (5)
I believe the answer is:
alibi
'story' is the definition.
(thesaurus)
'impromptu one with no depth' is the wordplay.
'impromptu' becomes 'adlib' ('ad lib' can be a synonym of 'impromptu').
'one' becomes 'i' (Roman numeral).
'with no' indicates named letters should be taken away.
'depth' becomes 'd' (abbreviation in geometry).
'adlib'+'i'='adlibi'
'adlibi' with 'd' taken out is 'ALIBI'.
'is' acts as a link.
(Other definitions for alibi that I've seen before include "Evidence that the accused could not have been guilty" , "Excuse for failure" , "that gets him off the hook?" , "Defence against charge - was seen elsewhere" , "Timely excuse" .)