Even better is that when you take that one city, you're guilty of 'genocide' and the rest of the world immediately turns against you... Even before the whole concept of genocide is invented! I did this recently to take full control of Africa on the world map... Didn't realize how severe the penalty would be - literally every civ in the world denounced me within a few turns, slashed luxury payments in half and ofc, last but not least, refused to sign RAs just as I got Education researched with 2000 gold in the bank.
So, lesson for all the cool kids out there: before you take the last city of an AI civ, be sure to open the Ingame Editor and grant them a nice resort in the arctic ice (somewhere where the sun doesn't shine - how fitting), then make peace and gift them that city. 10 turns later your vermin problem will be over with no adverse repercussions... Trust me, genocide has never felt better!