The Group B game in the French city of Lens kicks off at 14:00 BST on Thursday, 16 June.
The BBC will also broadcast Wales' game against Slovakia and Northern Ireland's Group C matches against Poland and world champions Germany.
The Group E matches between the Republic of Ireland and Sweden and Belgium against Italy will also be shown live on the BBC.
The BBC's head of TV sport Philip Bernie said the BBC had got "the most talked about match of the draw" in England v Wales, and will also show the opening matches for Wales and Northern Ireland in what is "such a special year for the home nations".
"Euro 2016 promises to be a standout event in a busy sporting summer on the BBC and we will bring audiences closer to the action than ever before with our most extensive coverage," Bernie added.
"We expect record numbers to enjoy our comprehensive range of content and catch-up services across TV, radio, online, digital, social media and the iPlayer."
Saturday, 11 June:
Albania v Switzerland, 14:00
Wales v Slovakia, 17:00
Sunday, 12 June:
Poland v Northern Ireland, 17:00
Germany v Ukraine, 20:00
Monday, 13 June
Republic of Ireland v Sweden, 17:00
Belgium v Italy, 20:00
Tuesday, 14 June
Portugal v Iceland, 20:00
Wednesday, 15 June
Russia v Slovakia, 14:00
Thursday, 16 June
England v Wales, 14:00
Friday, 17 June
Czech Republic v Croatia, 17:00
Saturday, 18 June
Iceland v Hungary, 17:00
Portugal v Austria, 20:00
Sunday, 19 June
Romania v Albania, 20:00
Switzerland v France, 20:00
Tuesday, 21 June
Ukraine v Poland, 17:00
Northern Ireland v Germany, 17:00
Wednesday, 22 June
Iceland v Austria, 17:00
Hungary v Portugal, 17:00
BBC has the second, third, sixth and eighth picks in the round of 16; first and second pick of the quarter-finals; second pick of the semi-finals, with both BBC and ITV showing the final live on Sunday, 10 July.
The Euro 2016 match between England and Wales will be shown live on the BBC.