Ok, that makes sense.
I am going to use these pages as a reference:
By definition, an honours degree in Scotland "is awarded for students who have completed four years at university - two years at sub-honours level, studying a variety of different subjects, and two years at honours level studying one subject in depth, usually including a dissertation in the final year." (This is taken from Wikipedia, which is open source -- so I'm allowed to copy it here.)
The following page tells you about the classifications of honours degrees:
First Class Honours means that you were in about the top 10% of your class. A 3.4 GPA is probably close to that (depending on where you go to school), but it likely falls just short, putting you in the equivalent of "Second Class Honours". (Which is then broken into upper and lower second class... see below.)
To be even more specific, you can look at this reference:
It says that last year, 12% had First Class, while 47% were in the Upper Second Class.
So to answer your question, I would say that your American 3.4 GPA is equivalent to Upper Second Class Honours in the UK.
I hope this helps,