I use the DS1307 extensively (Gawd knows why because today it's seriously overpriced for what it is - time to look around for something better and cheaper) without problems... but I ALWAYS have a Battery (CR2032)... it seems pointless having Timekeeper hardware if you allow it to lose its settings each time you power-off. I'll confirm that it's kinda hit or miss if it'll work without a Battery and you leave VBAT floating. If you really don't need a Battery, then a couple of Resistors as a potential divider across Vss and Vdd to give you about 3v... a quote from the Datasheet...

"VBAT – Battery input for any standard 3 volt lithium cell or other energy source. Battery voltage must be held between 2.0 and 3.5 volts for proper operation. The nominal write protect trip point voltage at which access to the real time clock and user RAM is denied is set by the internal circuitry as 1.25 x VBAT nominal. A Lithium battery with 48 mAhr. or greater will back up the DS1307/DS1308 for more than 10 years in the absence of power at 25 degrees C."