Public Eating and drink is not permited during the month of Ramadan in UAE. If you happen to be in Dubai During Ramadan you need not worry much for there are plenty of restaurants that respectfully open to serve non-Muslims during Ramadan.

Oud Metha to Deira
Bistro Madeleine: Heartwarming French food with great service.
InterContinental, Festival City (04 701 1111). Open daily 6am-11.30pm

Café Arabesque: Luxurious Levantine spot.
Park Hyatt Dubai, Deira (04 317 2222). Open daily 6.30am-midnight
Casa Mia: Classic Italian with a great cellar.
Le Méridien Dubai, Garhoud (04 217 0000). Open daily 12.30pm-3.45pm, 7.30pm-11.30pm
Epicure: Fantastic international café.
Desert Palm, Al Awir Road (04 602 9333). Open daily 7am-10.30pm
Kisaku: Tasty mid-range Japanese food.
Al Khaleej Palace Hotel, Deira (04 223 1000). Open daily 12.30pm-2.45pm, 6.30pm-11pm
Market Café: A lovely deli spot.

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Dubai Desert Safari