Queen Elizabeth II is well known as an affectionate breeder of Welsh Pembroke Corgi dogs; most of the pups she keeps from litters born to her dogs are female. Her sister Princess Margaret had a male Miniature Long-Haired Dachshund called Pipkin who seemed to love the Corgis in quite a different way from their owner -- a litter of seven Dachsund/Corgi crosses was born to one of the Queen's dogs. However, Elizabeth found them appealing enough to christen the type "Dorgi" and arrange for more to be born. (This is occasionally spelled "Dorgie".) Photographer Norman Parkinson was apparently bold enough to ask how the short-legged Miniature Dachshund could manage to mate with the comparatively taller Corgi and was told by the Queen, "Oh, it's quite simple. We have a little brick."
The one illustration I've been able to find shows a dog which resembles the dachshund parent more in shape, though its ears are not so large and floppy and its muzzle is a bit more pointed than most dachshunds.
"In 1975 the Kennel Club of England commissioned a portrait of its patron, Queen Elizabeth II. This work shows Her Majesty surrounded by some of her dogs . . . Centred in the portrait is the cross-bred 'Tinker' whose dam was a Pembroke Corgi, his sire a Dachshund--a breed the Royal Family calls the "Windsor Dorgi." Desmond Morris' Dogs says that the Kennel Club pointed out that there was not a single cross-breed dog depicted in any of the portraits hanging in the club, but Her Majesty said that if Tinker were not allowed in the portrait, then she would not sit for it, so the club relented. (It's good to be the Queen!) Tinker, Berry, Brandy, Cider, Harris, Pickles, and Chipper and others of the same mix lived happy, pampered lives among their pure-bred relatives, though the Queen's enthusiasm has not really led to a rush by other breeders to make more Dachshund-Corgi crosses. The Secretary of the Kennel Club at the time of the portrait conflict is supposed to have remarked, "The Dachshund was evolved to chase badgers down holes and Corgis to round up cattle. If anyone loses a herd of cattle down a badger hole, these are just the dogs to get them out."
Sources:
Morris, Desmond. Dogs: The Ultimate Dictionary of Over 1,000 Breeds. North
Pomfret, Vermont: Trafalgar Square Publishing, 2001.
http://www.ckc.ca/scoop/breed/default-refresh.asp?load=160
http://ladydiana.portal.dk3.com/article.php?sid=1286
http://home.austin.rr.com/fatdog/general.html
http://blogsisters.blogspot.com/
http://www.royalty.nu/Europe/England/Windsor/ElizabethII.html
http://www.guardian.co.uk/weekend/story/0,3605,724638,00.html
http://bunny-butt.tripod.com/fam_dogs.htm