Bill got tuberculosis around August 1st (3)
I believe the answer is:
tab
'bill' is the definition.
(a tab is a bill at a bar)
'tuberculosis around august 1st' is the wordplay.
'tuberculosis' becomes 'TB' (synonyms).
'around' means one lot of letters goes inside another.
'august 1st' becomes 'a' (1st letter of 'august').
'tb' placed around 'a' is 'TAB'.
'got' is the link.
(Other definitions for tab that I've seen before include "Running bar bill" , "Check" , "Text indent key" , "Tag" , "Flap; meal bill" .)