Another route, and one which DOES allow you to capture the other civilization's base culture. Note that this method almost requires you to target a city which borders on your own civ.How to steal cities from another civilization without war declaration?
(1) Build a considerable number of your best available hidden nationality units, particularly those along the Thief/Rogue/Burglar line. Optimize their promotions for infiltration and disguise - the idea is to make them as close to impossible for the opponent to spot or investigate.
(2) Send these HN units to the target city, and once they arrive, infiltrate the city. If the infiltration is successful, your Thief/Rogue/Burglar/whatever will be returned to your capital and you will get a boost to your espionage points against the opposing civ. You should immediately send any returned criminals back to the target city and do it again, and again, and again... The concept here is to build up your espionage total against the target as quickly as possible and to continue building it.
(3) Once you think you have enough criminals on this treadmill, build several spies. First promotion should be Loyalty (so any captured spies won't give you a diplomatic hit), second should be Mobility (to take full advantage of roads and such to get the spy to the target as quickly as possible). After those promotions, use your best judgement on subsequent promos. Those two are the important ones for the spies.
(4) As each spy comes on-line, send it to target city. Unlike your criminals, you will be parking your spies in the city to build up to the maximum bonus. Once your spy reaches his/her maximum preparation bonus, use him/her to spread your culture to the city. Again, your spy should return to your capital - immediately send him/her back to the target city and repeat.
In the case of a well-established target city, you'll probably end up with five to ten spies (or even more), with four to six criminals per spy, all on these complementary treadmills, building up and spending espionage points and putting more and more of your culture into the target city. After a while, the city will begin suffering rebellions and it will eventually revolt and join your empire, most likely without the other civ declaring war.
The advantage of using your criminals in this manner rather than parking them in the target city is that, once the city does flip, you haven't had criminals parked in the city for ages, building up crime, so it's much simpler to bring any unrest/crime back under control. (Building up crime in an enemy city is much more a tactic for when you intend to destroy the city, not capture it. Cleaning up the mess afterwards when you use a crime offensive can be a real pain.) Also, once the city flips, you have all those spies and criminals available to switch to your next target city...