In a medium bowl, mix balsamic vinegar, rosemary, garlic and herbes de Provence. Place beetroot in the mixture, and marinate at least 20 minutes.
Preheat an outdoor barbecue for high heat, and lightly oil grate.
Place the beetroot and marinade mixture on a piece of foil large enough to wrap all ingredients, and seal tightly. Place the foil parcel on the prepared barbecue, and cook 25 minutes, or until beetroot is tender.
Remove beetroot from the packet, and place directly on the barbecue grate for 2 to 5 minutes before serving hot.