No of course the verbal contract is very important- naturally it is harder to prove than a written one- but that is only if the dealer denies having said this. Do you have any witness to this verbal contract?
Chances are if he is a reputable dealer, he will stand over his word- after all reputation is important. Have you gone back to him yet and asked him to repair the car, or is that what you want?
In the first instance, you should look to the dealer to either repair or replace the car. If he refuses, then I'd get onto my solicitor pronto.
BTW the fact that he is a dealer rather than a private individual is in your favour- if he was a private individual, really caveat emptor ( buyer beware) usually applies. But for a car dealer who is an expert, he should have to stand over the goods he sells.