Alexion's offer of $6.30 a share cash is a 72.6% premium to Achillion's closing price of $3.65 on Tuesday.
Achillion shares were up nearly 75% to $6.39 in trading Wednesday; Alexion shares declined 1.33% to $103.42.
Alexion is seeking to expand its pipeline of rare disease treatments through Achillion, a clinical-stage biopharmaceutical company.
The deal is expected to close in the first half of 2020, subject to approvals including Achillion shareholders, the companies said in a press release.
"Alexion is an established leader in developing medicines for complement-mediated diseases, and we look forward to working together to accelerate our objective of bringing novel therapies to patients as quickly as possible and ensuring that the broad promise of this approach is fully realized," said Joe Truitt, Achillion's president and CEO.