Well... about the text... I think it should be "Il mio cuore do a voi" or (less probable) "mi do a voi con (tutto) il cuore". I would prefer the first possibility, in which you could have "I give you my heart" or, to emphasize it more "I give my heart to you", as you indicated. The second example could only mean "I give myself to you with (all) my heart", but it is of course another type of sentence.
The best solution could be to ask to change the original text, eliminating the word "mi" before the word "do".
I have a doubt in the translation of "voi" with "dig". If there are more people maybe it could be better "er", unless "voi" is a polite (even if old) form for the 2nd pers. sing.
Hoping that I could help you in a way, let's see of course if there are other opinions
Thank you very much, everybody! Now, one more request: everyone seems to agree that the word "mi" does not fit in this text or is strange. Maybe we could simply remove it? Is there anyone of you, Italian experts, who could acctually do that?
And, lilian canale, what about "till dig" or "till er"? Would it be okay to change or do you have another opinion on it (please read the ads above)?