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The Gregorian calendar has no year 0, however this has nothing to do with when decades begin or end.  Decades are named this way: the 60's, the 70's, the 80's, the 90's, etc....  The 60's began in 1960 and ended in 1969.  The year 1970 could no longer be described as being part of the 60's, because it clearly has a 70 in it.  Decades are named according to the number in the tens place and not according the beginning of the Gregorian calendar.  

Therefore the current decade is about to end in a few days.