The rise was boosted by sales of private cars, which were up by more than 20% last month compared with June last year, the Society of Motor Manufacturers and Traders said.
New car registrations for the year passed the one million mark in June, increasing by 10% compared with the same period in 2012 to 1,163,623.
Motorists switching to more fuel-efficient cars helped reduce average new car CO2 emissions by 3.7% on a year ago.
SMMT interim chief executive Mike Baunton said: "Boosted by consistently strong private demand and further growth in June, half-year new car registrations have topped the one million mark.
"June secured the 16th month of consecutive growth, a clear indicator that manufacturers and dealers are delivering desirable new products with tangible cost savings from the latest fuel-efficient technology coupled to a wide variety of competitive finance offers.
"While there are still potential challenges ahead, recent robust growth suggests that the market is on course to perform well ahead of 2012 levels."
The SMMT said there had now been six successive quarters of growth, with the 13% in the three months to June the strongest quarterly rise since the autumn of 2011.
The business organisation added that the UK continues to buck new car market trends in other European countries, with buyers returning after a "sustained" hold-off from buying new vehicles, partly because of enticing finance deals.