Expedition to the French metropolis goes around a river (8)
I believe the answer is:
'expedition' is the definition.
(I've seen this before)
'to the french metropolis goes around a river' is the wordplay.
'to the french' becomes 'a la' ('to the' in French).
'metropolis' becomes 'city' (a metropolis is a large city).
'goes around' means one lot of letters goes inside another.
'a river' becomes 'r' (abbreviation for river).
'city' placed around 'r' is 'crity'.
(Other definitions for alacrity that I've seen before include "Showing brisk readiness" , "prompt response" , "Promptness, eagerness" , "Lively enthusiasm" , "Expedition" .)