In the fall, they could just park the train for an hour, but in the spring, wherever a train is, it will be an hour late for the rest of the journey!
From their national timetable (for 2012, but I assume that the policy hasn’t changed):
Unfortunately, being an hour behind schedule is not an uncommon occurrence on long-haul Amtrak trains, so I suppose it makes sense that they just treat the spring time change like any other delay.
[emphasis added]
… And no one will notice the difference from any other day of the year!
OK, they’re not always late, and not usually very late, but it’s not unusual either. Which is better than getting into accidents trying to maintain a to-the-second itinerary. It’s not like any other mode of long-distance travel is always on time, either.
Another factor is that Amtrak has wiggle room (for want of a better term) built into their schedules. On my trip from Chicago to Seattle last year, we were 41 minutes late leaving Havre MT; by the time we reached Shelby, 108 miles further on, we were dead on time. And we got into Seattle 39 minutes early.
Seeing as the railroads pretty much invented time zones and accurate time keeping, Amtrak probably handles the change gracefully.
Not that they pay much attention to time in the first place… 
On a related topic, it is also worth noting that their online published schedules do not account for the fact that Arizona does not participate in daylight savings time. The Southwest Chief will nominally be an hour early into Flagstaff in the summer for this reason, or just slightly early in practice. The Arizona stations do post and quote the corrected times but it made for a long wait for my ride on one occasion.
The biggest challenge which Amtrak faces in trying to run on schedule is that, on the vast majority of their routes (the Northeast Corridor being a notable exception), they’re running their trains on someone else’s track. The tracks are owned by freight railroads (such as BNSF, Union Pacific, and CSX), and those railroads will often give schedule priority to their own freight trains, bumping the Amtrak train to a lower priority.