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Both Walter and outfielder Kevin Jordan were recovering Tuesday in an Atlanta hospital one day after the transplant was performed.
"For us, it's almost like it's been divine intervention," Jordan's father Keith told the Associated Press.
Dr. Kenneth Newell, the lead surgeon on the team that removed Walter's kidney, said in a statement that he expects Walter and Jordan to recover fully.
The school says the recovery time for both the 42-year-old Walter and Jordan is expected to be several months. Keith Jordan said he isn't worrying about when his son, a 19th-round draft pick of the New York Yankees last June, may return to the field.
"One of the things we do know for Kevin is, he's going to want to go do stuff right away," Keith Jordan said. "He's going to have to take care of himself. ... His intention is to get back on the field, so I'm sure he's going to do whatever it takes to do that."
He was diagnosed last April with ANCA vasculitis, a type of autoimmune swelling disorder caused by abnormal antibodies. He wound up on dialysis -- three days a week at first, and then daily. Family members were tested to see if any were a possible match for a transplant, and Walter was tested in December after it was determined that his relatives weren't compatible.
Walter found out Jan. 28, that he was a match. He told the team three days later, and said the players greeted the news with "stunned silence followed by a round of applause."