We found a very nice translation into Russian of the beginning of the song done by one of the workers of some Russian translation company.
So, in the original it goes like this:
Sleigh bells ring, are you listening,
In the lane, snow is glistening
A beautiful sight,
We're happy tonight.
Walking in a winter wonderland.
And here is English to Russian translation of the song:
Слышишь, колокольчик в санях звенит,
На дороге снег блестит,
Красота - не передам!
Cчастливы сегодня вечером,
Гуляя в зимней сказке.
I think the translation was made pretty-pretty well.
And for those of you who wants to try to sing it in Russian, here is the transliteration (Russian phonetic translation):
Slishish', kalakol'chik v sanyah zvinit,
Na darogi sneg blestit,
Krasata - ni peredam!
Shastlivi segodnya vecheram,
Gulaya v zimnei skazke.