William & Kate Enjoying Private Mustique Babymoon

02/07/2013 at 02:10 PM EST

William and Kate - Inside Their Private Vacation in Mustique
The Duke and Duchess of Cambridge (inset) visited Aurora House, Mustique
IPHOTO; Inset: Splash News Online
The The Duke and Duchess of Cambridge are keeping a low profile on their "babymoon" to the idyllic island of Mustique.

The couple – expecting their first baby in July – are staying at a five-bedroom villa set in lush gardens, which normally costs an eye-popping $25,000 a week. Also enjoying paradise are Kate's parents Carole and Mike and siblings Pippa and James.

On previous visits, William, 30, has stopped by to grab a drink and play the piano at the bar at Firefly, and together the couple have participated in Sunday night Karaoke at Basil's Bar.

But on this visit, the couple are mostly keeping to themselves, and they have good reason.

With a staff of six (including a chef on standby), a 60-foot infinity pool and a media room and guest cottage, the villa has everything they need. And, best of all for the private couple, it's completely hidden from the main road and with just the tips of the pitched roof line visible from the beach down below.

The stunning beach, which many believe to be the finest on the island, is just a short walk down a private path from the house. On their past trips the sporty pair made several trips a day down that path to stroll on the beach, take quick dips in the ocean and also to snorkel well past the rocky outcrops that border the cove. The beach is still public however, and the couple save their sunbathing for the villa's expansive pool deck. 

From the lounge chairs, pregnant Kate, 31, will be able to look along the surface of an infinity pool and out to the other islands of the Grenadines spread out in the ocean below.

Mustique, a favorite of the Middletons for years, and made famous by William's great aunt Princess Margaret and her friends in the 70s, ensures the privacy of its high-end visitors. A local tells PEOPLE, "The island is locked-down – that always means some royalty is here."

When they return, William is due back at his north Wales base, from where he flies search-and-rescue helicopters, while Kate is scheduled to visit a drug treatment center in London that is run by one of the charities she supports, Action on Addiction.

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