The male patient, 54, was "doing well under the circumstances" after the 15-hour operation on July 25-26, a spokeswoman for the clinic at the Technical University in the southern city of Munich said.
The amputee, who had lived without arms for six years since the accident, consulted the 40-member team at the university's Rechts der Isar Clinic after two failed attempts to use artificial prostheses.
"The man required round-the-clock assistance -- a condition he wanted to change as quickly as possible," the clinic said in a statement.
The head of the transplant team, Christoph Hoehnke, told reporters he was deeply moved as the man's wife went to his bedside after the operation and instinctively reached for his hands.
"They look just like they used to," she said, according to Hoehnke.
The patient was in a good condition but it could take two years before he "really has feeling in his fingertips again" because the transplanted nerves must still grow, the clinic spokeswoman said.
The facility has a decades-old unit for microsurgery and replantation surgery, with a speciality in interdisciplinary operations it said was essential for a procedure of this complexity.
Professor Hans-Guenther Machens had prepared the transplant since he became the clinic's director in December.
Doctors said suppressing the man's immune system so it would not reject the new limbs was a key concern.
Another challenge was finding a donor who matched the patient's sex, age, skin colour, size and blood type.
Five teams working in two operating rooms gathered at 10:00 pm the night of the operation, divided between the patient and the donor, who had died only hours before.
The first step was to expose the muscle, nerves and blood vessels to be connected. Before the bones of the donor could be cut, blood vessels in his arms were filled with a cooled preservation solution.
Both arms were then removed exactly at the point matching the patient's arm stumps. First the bones were joined, then arteries and veins to ensure blood circulation as quickly as possible.
"The arms quickly resumed their rosy colour," the spokeswoman said.
The surgeons then attached the muscles and tendons, then the nerves and finally the skin.
The doctors will now monitor how the wounds heal and whether infections, side effects from medication or any other rejection by the immune system occurs.
But they said they were pleased with the progress of the patient, who is receiving physical therapy as well as psychological counselling.
The clinic said that hand and lower arm transplants were still rare and that the Munich operation, by attaching an elbow joint as well as an upper arm, posed a greater challenge for the immune and circulatory systems.