From next the 1 june ours society adottera' the Euro as account currency.
Interesting that Babelfish translated (sort of) all of the sentence except the verb. Adottera' is the future tense of "adotter" which is "adopt" Thus the original sentence says, in better English, "Beginning June 1 our society (could this mean company?) will adopt the Euro as coinage of account" (presumably an Italin idiom for .... hum, what do we call it over here?)
Denominera' again is a future tense "will be named" disporra' is "will dispose"
I agree that it looks like they are saying all their business will be in Euros from now on. Your agent is the one who will have to deal with this, I should think.