In one word, yes. I think saying "the soviets" is more accurately "a Soviet scientist/doctor" because he wasn't doing it directly in service to the government.

Vladimir Demikhov is the name of the scientist who performed this experiment, circa the 1950s. Demikhov was a graduate of Moscow State University, then drafted into the war and on the front lines in China.

How did he get to the point of grafting a dog's head onto another dog? Demikhov was a pioneer in transplant surgery, as well as creating the first artificial heart. Some argue that he should be recognized more for his contributions to actual medical science than his seemingly-insane dog experiments, but I digress. The dog experiments did not contribute to medical science--while head transplants have been attempted since the early 20th century, they have yet to be successful in animals or tried in humans (though a couple years ago a Russian man did volunteer to undergo the procedure, he backed out).

He began the experiments in 1954 and it was kept secret, as expected with the Soviet Union, until 1959 when news spread around the world. By this point, he had attempted this twenty-four times. This was not the most successful attempt, but since it was published in Life magazine it attracted attention in the United States and throughout the world.

A smaller dog's head and front legs (with it, part of the chest, including the heart and lungs) were grafted onto a fully grown German Shepherd. This was to keep the smaller dog alive until the procedure was done. This particular dog/specimen/amalgamation only lived for four days and died from an injury to the neck, while his longest had lived for 29.

So technically, the Soviets made a number of two headed dogs, with one being world famous.