To take benefit of the best prices you will need to prove you are a leisure traveller, on the way to the airport, so you will need to give either your flight details or your holiday booking reference. I don't think you will need to give this when you arrive at weymouth station but you may be asked to provide them on your journey from the station to the airport.
Going to gatwick airport by train is great, you can take a nap, read a book (even at night), play travel scrabble, or just gaze out of the window... The other restriction to buying these cheap tickets from the station to the airport is that there must be at least two grown ups travelling together. But another great advantage is that children go free. All children under 5 go free, and then one child over five but under sixteen per grown up travelling on the train to the airport.
be sure to leave enough time to get to the airport early enough to eat, shop etc, so check the times of trains from weymouth station before you go. If you end up with spare time at the airport you can always read a book or play travel scrabble again! As you have bought yr train ticket in advance, you don't need to worry about the opening hours of the station.
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