We show that the number S_n of rounds required to spread a rumor in a complete graph with n nodes is very closely described by log_2(n) plus (1/n) times the completion time of the coupon collector process. This in particular gives very precise bounds for the expected runtime of the process, namely floor(log_2(n)) + ln(n) - 1.116 <= E[S_n] <= ceil(log_2(n)) + ln(n) + 2.765 + o(1).
This is joint work with Benjamin Doerr.