train from paris to salzburg

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Fastest way is 7 hours and 19 minutes with a single connection in Stuttgart, but that is a 9 minute connection time. Some others in 8 hours with more comfortable connection times.

It would likely be Paris (Gare de l'Est) with changes in Manheim and Munich; or in Stuttgart. Minimum 8 hours. Probably longer. This is one of those situations where it might be quicker to fly into Munich and then catch the train from there, or if you're lucky, get a flight from Paris to Salzburg.

Flights from Paris to Munich are operated several times a day and can be as cheap as $80 if booked well in advance. From Munich take an express train to Salzburg. This will save you more than 5 hours of travel time in total.

Check for best airfares: https://www.kayak.com/flights

I'd rather take the local Meridian train than the express train. From MUC take the S8 to Munich East Station and change there. That connection is no slower (sometimes even a few minutes faster) than the one via the main station and can be used with a Bayern Ticket for €25 for one traveller or €32 for two (valid from 9am on workdays, all day on weekends; each additional traveller up to five: +7€). A flexible long-distance train ticket via the main station would cost €40-44 pp. There are cheaper train-specific tickets, but if you can't reach your booked train due to flight delay, such a ticket will be worthless. All details on www.bahn.com