Take a taxi from Ben-Gurion Airport’s taxi dispatcher station. The cost should be around NIS 160 (approximately $40 U.S. dollars), and the ride should take around 30-40 minutes.
If you are travelling light, bus #249 travels to Rehovot's central bus station from the airport. This is not, however, an express bus and the drive will take approximately 1:15 hours. The cost is about US $3.00. Once at Rehovot's central bus station, there are a multitude of local buses which go to the Weizmann Institute (ask at the information window), or you can take a taxi from the taxi stand directly across the street from the bus station.
You can also get to the Weizmann Institute by taking the train from the airport. Change at the Tel Aviv Hagana stop for the train to Rehovot. The train station in Rehovot is about a 5-minute walk from the Weizmann Institute.