An exam in the morning? That’s unethical! (6)

I believe the answer is:
amoral
'unethical' is the definition.
(I've seen this before)
'an exam in the morning?' is the wordplay.
I cannot quite understand how this works, but
'an' could be 'a' and 'a' is present in the answer.
'exam' could be 'oral' (spoken academic test) and 'oral' is present in the answer.
A single letter 'm' remains which might be clued in a way I don't understand.
This may be the basis of the clue (or it may be nonsense).
'that's' is the link.
Can you help me to learn more?
(Other definitions for amoral that I've seen before include "Lacking principles" , "Ignoring what's right and wrong" , "outrageous!" , "Unconcerned with correctness" , "Without an ethical code" .)