time = (hour * 100) + minute;
On large parts of this planet, there are 60 minutes in an hour.
if ( (time >0800) && (time <+ 2030) ){digitalWrite (1, HIGH);} else {digitalWrite (1, LOW);}
Supposing that hour had held a value of 9 and minute had held a value of 20, time should contain the value of 930. This is greater than 800 and less than 2030, so you set digital pin 1 to HIGH. Well, that just screwed up your serial communications. Digital pins 0 and 1 are the serial port on the non-Mega Arduinos, and one of 4 serial ports on the Mega. Those pins can not ALSO be used for GPIO.
Im Still Not Having Any Luck With The RTC...
There Is No Luck Required. Either You Can Read The Time From The RTC Or You Can't. wHICH iS iT?
If you can't get the time, why are you trying to use it? If you can get the time, what is the problem?
year = RTC.get(DS1307_YR,false);
month = RTC.get(DS1307_MTH,false);
date = RTC.get(DS1307_DATE,false);
hour = RTC.get(DS1307_HR,true);
minute = RTC.get(DS1307_MIN,false);
second = RTC.get(DS1307_SEC,false);
Get the year, month, and date from the last time those values were fetched from the clock, then fetch new values, and get the hour, minute, and second values. Makes sense. Not.
Is the time on your RTC set? You are using hour and minute, but printing only day, month, and year. Why not print the values that you actually use?