Severus Snape was already a Death Eater.
He approached Dumbledore after having already told Voldemort about the prophecy.
“What request could a Death Eater make of me?”
“The — the prophecy . . . the prediction . . . Trelawney . . .”
“Ah, yes,” said Dumbledore. “How much did you relay to Lord Voldemort?”
“Everything — everything I heard!” said Snape. “That is why —
it is for that reason — he thinks it means Lily Evans!”
“The prophecy did not refer to a woman,” said Dumbledore. “It
spoke of a boy born at the end of July —”
“You know what I mean! He thinks it means her son, he is going to
hunt her down — kill them all —”
“If she means so much to you,” said Dumbledore, “surely Lord
Voldemort will spare her? Could you not ask for mercy for the mother,
in exchange for the son?”
“I have — I have asked him —”
“You disgust me,” said Dumbledore, and Harry had never heard so much
contempt in his voice. Snape seemed to shrink a little. “You do not
care, then, about the deaths of her husband and child? They can die,
as long as you have what you want?”
“Hide them all, then,” he croaked. “Keep her — them — safe.
“And what will you give me in return, Severus?”
“In — in return?” Snape gaped at Dumbledore, and Harry expected
him to protest, but after a long moment he said, “Anything.”
Early we see Lily say,
You and your precious little Death Eater friends — you see, you don’t even deny it! You don’t even deny that’s what you’re all aiming to be! You can’t wait to join You-Know-Who, can you?