The latest increase means the cost of filling up an average family car with petrol is now £5.64 higher than it was at the beginning of March.
But the motoring organisation says drivers are still paying less for their fuel than they were a year ago.
According to RAC Fuel Watch, the average price of petrol increased by 1.5p a litre to 112.17p in June. The price of diesel, meanwhile, rose by 1.66p to 112.39p per litre.
That means it now costs £61.69 to fill up a typical family car with petrol compared to £60.88 at the beginning of June and £56.05 four months ago when the average price was 101.91p a litre.
Diesel, meanwhile, now typically costs 11p per litre more than it did in early March, meaning a comparable fill-up will now set motorists back £61.81.
The cost is £5.96 more than at the beginning of March and 91p higher than a month ago.
Commenting on the latest rise, the motoring organisation’s fuel spokesman, Simon Williams, said: “June was another bad month for motorists with the price of petrol going up again. While it was only a penny and a half it makes for a rise of more than 10p since the start of March.
“Filling up with unleaded is now £5.64 more expensive, which is enough to make an unpleasant dent in household budgets up and down the country, especially for those who have more than one car or need to fill up regularly.”
Despite June’s increase motorists are still paying about 5p a litre less for petrol and 8.5p less for diesel than they were 12 months ago, Mr Williams added.
Wholesale fuel prices, meanwhile, only increased slightly after the fall in the value of the pound following Britain’s decision to leave the European Union, he said. That, Mr Williams added, was due to oil prices dropping at the same time.
Looking ahead, Mr Williams said OPEC’s over production was likely to keep the price of oil at around $50 per barrel.
Prices could even fall, he added, once production issues have been resolved.