This is OK in AE, but not in BE (= starting on the 1st of June.). Also it is ambiguous as it is not clear whether "starting the first of June" qualifies "I am available" or "the internship"I am available for the internship starting the first of June.
Let me emphasise how important it is to capitalise the letter "I" - applications are regularly dismissed for failure to capitalise "I"."I am available for the internship from 1st June to 3rd August"
"I am available for the internship from the first of June."