A Night to Remember (1958)
Directed by Roy Ward Baker, written by Eric Ambler based on Walter Lord's book.
A Night to Remember was obviously... well, Titanic of its time, if you excuse me for using this extremely clumsy comparison.
The movie makers managed to contact a lot of Titanic survivors, and made everything to make sure the movie felt as authentic as possible. The movie making involved building a detailed 10-meter model of the ship.
The movie mostly concentrates on the accident, leaving the leave and the travel to lesser focus. The accident is very well detailed, aside of the slightly accelerated sense in the very end.
ANItR was, to me, the movie that made me to realize the true horribleness of the whole event. I had seen the National Geography documentary and local science mag's photographs from Robert Ballard's discovery. That was when I was a kid. Nothing scary. Just black and white photos and numbers and dates and pictures of a rusting wreck. Now, this movie on the other hand... I could feel the terror creeping among the passengers.
There are some mistakes in the movie that even amateurs can spot: For example, the film begins with christening of the ship, and Titanic was never christened (the footage is from christening of Queen Elizabeth). And, of course, the ship sinks intact (but the fact that the ship broke in two was not confirmed until the wreck was found!)
Obviously, many later Titanic films are in debt to this movie; The 1997 Titanic has a couple of scenes that are clearly influenced by it, or are almost similar. (In fact, so similar that I can't remember in which movie that was. Either the makers borrowed the scenes or both made startlingly accurate depiction of eyewitness account =)
Now, I only wish I could find this on a DVD. The copy I have is actually a CD-ROM with a video as a QuickTime file; It has an UI that has all sorts of extras, including something not found in many DVDs - a full word search of movie dialogue.