Yes, it should be truncated
Per the spell, there are no limitations to what you can say. This is contrasted by other spells, like Sending, that have a strict limit on what you can say. But, even though Message doesn't have a limit, you are still limited by the other game rules. Just because you are not limited by the spell itself (other than the 1 round duration) does not mean you can break other rules.
What do the rules say?
Page 181 of the PHB describes a round as lasting 6 seconds. On page 190, it goes on to explain (under "Other Activity on Your Turn")
You can communicate however you are able, through brief utterances and gestures, as you take your turn...
The DM might require you to use an action for any of these activities when it needs special care....
What this means is that there is a precedence for communication being limited during combat. The former snippet of the quote is for doing things while communicating. What you are doing with Message is dedicating your entire action to speaking and casting, so you are not limited to "brief utterances", but you are limited by the concept of time in a round. If you can't say it in 6 seconds, it should be cut off.
Normally speaking, if you said the same message aloud as not part of the spell, it might also require an action to get it out (DM's discretion), since you are giving special attention to communication and not spell casting or attack rolls. However, with Message you are whispering the message as part of the spell, so it requires no extra time or effort.
Your message should be limited by the 6 seconds, no more or less. There is no precedence for a spell effect and a casting time being exclusively part of the round without any sort of indication that it is doing so: you use your action to cast and whisper the message. Not cast the spell then whisper.