There are 2 train companies operating 3 routes - R21;KBS 145, ICE28, R10. The cheapest train costs $59 and is 5h 13min.
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There are 4 ways to travel from Dessau to Lübeck. 4 buses, 3 trains, 1 flight and 1 by car are possible between the two areas. The by car from Dessau to Lübeck is the cheapest and fastest. The train operated by running from Gräfenhainichen to Lübeck is the most convenient.
It's important to check each mode when booking your ticket so that you can find the best method for you. The cost of each type varies differently, so be sure to use Faretrotter to keep an eye out!