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Travel from London to Warsaw by train. It takes just a night and a morning, with comfortable sleepers and couchettes available.
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Spend the night in Warsaw.
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Travel from Warsaw to Vilnius by train. There is a daily train service, leaving Warsaw Centralna at 07:20 and arriving Vilnius at 17:56 the same day, with a simple change of trains at Sestokai, see the section below. This service does not pass through any part of Belarus. You can double-check train times for your date of travel at DB BAHN - Verbindungen - Ihre Anfrage (English button upper right). Check the destination boards on the side of the coach you board, to make sure it's one of the through cars to Sestokai - other coaches in this train terminate at Suwalki, the stop before the frontier. Map of Vilnius, showing railway station.
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The only other option is a very long all-day or overnight bus journey from Warsaw to Vilnius, see eurolines.pl or Eurolines.lt. Not something that was necessary in communist times, when a good train service operated.!