Carefulness with which engineer approaches old Chaldean city (6)
I believe the answer is:
rigour
'carefulness' is the definition.
(rigour can mean being careful or thorough)
'engineer approaches old chaldean city' is the wordplay.
'engineer' becomes 'rig' (as a verb).
'approaches' means one lot of letters go next to another.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'chaldean city' becomes 'Ur' (city in the Hebrew bible).
'rig'+'o'+'ur'='RIGOUR'
'with which' acts as a link.
(Other definitions for rigour that I've seen before include "Strict exactitude" , "Hardship" , "Severity, harshness" , "Austerity - strictness" , "Trial" .)