The accuracy of your system depends on how good your main time base is. In the case of DS1307 it depends on whether you have chosen a proper crystal for your chip, and its capacitive loading.
see this note for details -- **broken link removed**
Of course your measurement method for the accuracy should also be very good. How are you testing the accuracy ?
is your change an increase or a decrease ? does it go faster or slower ?
you can improve the accuracy using guidelines from AppNote 58, or go the easier route and use ds3231
it all depends on what you mean by extremely accurate
upto 60 seconds in 6 months means accuracy/ stability of ~4 ppm. DS3231 datasheet claims to be able to do better than this with proper care in all matters.
i do not have personal experience with this ic however
This suggests you have access to mains AC. Is it reliable enough that you could use it as a timebase? Divide down 50 Hz (or 60 Hz) to get 1 Hz pulses, etc.