Interpretation of Train's song "Marry me"

Summary

This is a story of a man who saw a girl and then said with full assurance to himself, "she's the one!".

I

"Forever can never be long enough for me
Feel like I've had long enough with you
Forget the world now, we won't let them see
But there's one thing left to do"

(Line 1) He wants to marry the girl and spend the rest of their lifetime with each other,

(Line 2) It feels like they've been with each other long enough that they wouldn't know how it is to live without each other.

(Line 3) Don't mind everyone because they won't know the real story.

II
"Now that the weight has lifted
Love has surely shifted my way"

Verse II states that the barrier/wall/awkwardness when you're newly acquainted is gone(In short, they're done with the "getting-to-know-each-other" stage), and he's sure of his feelings for her.

III
"Marry me today and every day
Marry me if I ever get the nerve to say hello in this cafe
Say you will, say you will"

Verse III states that they've just met. He fell in love at first sight. And if he gets the courage to make a move on her, he'll make sure he would never let her go.

IV
"Together can never be close enough for me
Feel like I am close enough to you
You wear white and I'll wear out the words 'I love you'
And 'you're beautiful'"

(Line 1) It's not enough that they in a gf/bf relationship, he knows that she's the one that he'll spend the rest of his life with.
(Line 2) He feels that they know each other enough to get married.

(Line 3,4) She'll wear a wedding dress and he'll say that he loves her and that she's beautiful over and over.

V
"Now that the wait is over
And love has finally show her my way"

Verse V states how he has been waiting for that moment that he'll see the woman he'll spend the rest of his life with, and how he is thankful that destiny brought her to him.

VI
"Promise me you'll always be happy by my side
I promise to sing to you when all the music dies"

(Line 1) He wants to assure that she loves her too.

(line 2) and if she does he promises to make her happy and love her in times when all the problems in the world falls on them(the music dies).