A loss of service in 2016 revealed that uninterrupted access to emergency calls was vulnerable to a single technical problem, regulator Ofcom said.
Its investigation concluded that Three did not act recklessly, but the issue could be reasonably be avoided.
The company said it had strengthened its systems.
Gaucho Rasmussen, Ofcom's enforcement and investigations director, said: "Telephone access to the emergency services is extremely important, because failures can have serious consequences for people's safety and wellbeing.
