How can I give power to Arduino Uno using a 12V 7.5 Ah Lead-Acid battery? I am trying to give power to my Arduino Board using a 12V 7.3 Ah Lead-Acid Battery but I am unable to do it.

I connected the battery to Arduino Uno (via DC Pin) and it worked for a couple of minutes but then, it automatically stopped working. I checked the voltage of the battery using a Multimeter from the same DC Pin which was connected with battery and board. The voltage was 12.62 volts. 
Then I thought my board might have been fried. So, I checked the board by connecting it with a 12V 2 Amp DC adapter and I found that my board was not fried and it was working fine with the Adaptor. The Adaptor was giving 12.48 volts voltage. Then, I again checked the board using battery but still it didn't work. I tried charging the battery via a charger but it still didn't work. I tried these steps multiple times but the results were always same.

I checked with my 12V DC Fan &amp; some other devices also and it works perfectly, but it doesn't work with my Arduino Board.
This indicates that my battery is in a good condition but not working with Arduino Board only. While Arduino is also in a good condition, working with 12V DC Adaptors but not with the Battery.

Please Help!
The input voltage on the DC pin is specified as 7 to 12V. Your battery may well be supplying more than that, causing the voltage regulator to overheat and shut down.


  The voltage was 12.62 volts.


That is out of spec for the Arduino Uno. When you operate out of specified voltages anything can happen, and that anything may not be very good.

You could get a buck converter to lower the voltage.

Certainly a Uno should be able to run for a long time on a 7 Ah battery (provided you are supplying voltages in range for it, preferably using a buck converter or similar to make sure that a fully-charged battery does not supply too high a voltage).