Proof of a base immorality around hideout (8)
I believe the answer is:
evidence
'proof' is the definition.
(I've seen this before)
'a base immorality around hideout' is the wordplay.
'a base' becomes 'e' (Euler's number - base of natural logarithms).
'immorality' becomes 'vice' (vice can mean immoral behaviour).
'around' is an insertion indicator.
'hideout' becomes 'den' (synonyms).
'vice' enclosing 'den' is 'vidence'.
'e'+'vidence'='EVIDENCE'
'of' acts as a link.
(Other definitions for evidence that I've seen before include "Testimony or outward sign" , "Testimony - verification" , "Proof; testimony" , "The basis for belief or disbelief" , "Info given in court" .)