It is half past two in the afternoon, and Robert Plant is on stage in the bowels of an ice-hockey arena in Nashville. He is toting a mug of tea, as he often does (throat-soothing shot of honey optional). Road crew and musicians mill about, tweaking instruments, sound quality and lighting configurations. A forklift truck and a crane idle in front of the stage.

In the middle of all this activity, the queen of American bluegrass, Alison Krauss, is singing Green Pastures, a traditional gospel song that was popularised by Emmylou Harris. Plant nods along appreciatively. Now he has his hands planted backwards on his hips. He is part Billy Connolly, part Rigsby from Rising Damp and all rock god. He stomps towards the dainty Krauss throwing scary monster shapes and she responds by pretending that she’s a little pony and trots round the amplifiers and drum-kit. Then they fall about laughing.

The last time Plant was on a stage was in December last year, front and centre at the much-ballyhooed Led Zeppelin concert at the O2 in London. But here in Nashville, he is participating in a musical project that is, in its own way, just as remarkable as that legendary reunion.

Raising Sand, the album Plant and Krauss recorded in Nashville last year, is a collection of covers of obscure country, folk, R&B, soul and defiantly genre-free songs. The originals were sung by a disparate bunch of artists, including the Everly Brothers, Tom Waits, Gene Clark of the Byrds, and even Plant himself – Please Read the Letter originally appeared on Walking into Clarksdale, the 1998 album Plant made with Led Zeppelin guitarist Jimmy Page. Most were suggested by the album’s producer, T. Bone Burnett – although Plant did put forward a few songs from his own personal jukebox, a 1958 Wurlitzer he says he bought “for 60 quid when I was a hippy”, filled with a choice selection of 50 singles he has collected over the years.

Enveloped by Burnett’s gently atmospheric production, and by the empathetic fusion of the voices of Plant and Krauss, Raising Sand has been a critical and commercial smash. Since its release last October this beautiful album of duets has sold more than two million copies and waltzed, unheralded, into the upper echelons of the Top 40 on both sides of the Atlantic. It has given Plant the best chart placing of his solo career and been a boon for Krauss, whose nine critically acclaimed albums have not reached such heights.

“It’s all about restraint,” Plant notes. “If you just go waaagh!” – and here he lets out that familiar Zeppelin-era shriek – “where do you go next?” He does have form in the field of restraint. In 1984, he and Page, under the name the Honeydrippers, released a mini-album that features covers of Fifties standards. The project (or “adventure” as Plant calls such things) was encouraged by Plant’s great friend Ahmet Ertegün, the legendary co-founder of Atlantic Records and the man who signed Led Zeppelin in 1968. Plant’s hushed version of Sea of Love, originally a hit in the UK for Marty Wilde, reached No. 8 in the British singles chart. Nonetheless, not many people noticed it was that bloke out of Led Zeppelin singing. So this is the other remarkable aspect of Raising Sand: at the age of 59, Robert “Percy” Plant is reinvented as a – whisper it – crooner.

With Krauss, he is half of quite a double act.

He is the hard-rock banshee of Seventies lore, the former frontman of the most extravagant and heavy blues-rock band of all time and the “Golden God” the director Cameron Crowe “borrowed” for his iconic rock star in Almost Famous. He memorably demanded that a paramour “squeeze my lemon till the juice runs down my leg” in the song Killing Floor and, post-Zeppelin and post-Honeydrippers, spent many years exploring his interest in the blues and north African and Islamic music.

She is the God-fearin’ Midwestern girl, the bastion of America’s alt-country music scene who was a champion fiddler as a child, who released her first bluegrass album at the age of 16 and who has won 21 Grammy awards. She’s 23 years his junior and polite and almost diffident next to his raucous (and often ribald) earthiness.

When an executive at the music channel VH1 – who was friendly with both artists – suggested they work together, both were intrigued. First they sang at a 2004 tribute to the blues pioneer Leadbelly at the Rock ’n’ Roll Hall of Fame in Cleveland. Then, once their schedules allowed, they got down to it.

Now, on the stage in this dark and cavernous room beneath the Sommet Centre, home of the Nashville Predators, Plant and Krauss – under the tutelage of Burnett, who is trebling up as musical director and guitarist – are rehearsing for their first tour. Such has been the success of Raising Sand, they’re scheduled to appear at venues that can hold up to 15,000 people; in the UK they’re playing Wembley Arena.

As Krauss sings Tom Waits’s aching Trampled Rose, Plant and Burnett watch intently. “The first idea, before we had any songs, was just the sound of their two voices,” Burnett tells me later. “Listening to each one of them sing was almost hypnotic – some kind of psychotropic drug. So I thought listening to both them singing together might be very powerful.”

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