Dial +590 xxx xxxxxx. It will be +590 590 xxxxxx for a French landline, and +590 690 xxxxxx for a French mobile. That will work on both sides of the island. If the number is listed as 0590 xxxxxx then drop the leading 0 and substitute with +590.

For example, if you wanted to call Il Nettuno, their number might be listed as 0590 877738. Dial +590 590 877738, which will work no matter if you're connected to a French-side carrier or a Dutch-side carrier.