Upvote:5
From 1993 to 2005, Amtrak's Sunset Limited operated between Los Angeles and Florida, initially to Miami and later cut back to Orlando. Hurricane Katrina in 2005 damaged the CSX railroad line between New Orleans and Jacksonville, and the train was suspended east of New Orleans. The chronology of service appears on the Wikipedia page.
Upvote:6
The Panama Canal Railway, which crosses the Isthmus, was built by U.S. interests in what became the Panama Canal Zone, an "unincorporated territory of the United States".
Upvote:8
I guess the first one could be the Pullman's promotional train of 1870. It started from Boston on May 23rd, 1870, went to San Francisco and then back to Boston, completing the round-trip on July 1, 1870. It was not a regular line. Specifically built for the trip, it took 130 passengers, mostly wealthy members of the Boston Board of Trade with families.
http://cprr.org/Museum/Trans-Continental/index.html - the newspaper published aboard this train
Edit: Obviously, it had stops on the way!
Edit2: I understand the question is about the single train, and not concerned if the train continuously went all the way on the tracks (without a ferry).