I'd be interested in the success stories like above, especially a year or so afterwards. But that said...here's the opposing view.
From
here
Warning and caution: Some units will generate lethal amounts of AC power whenever the input power cord is unplugged-after all, this is their primary function. There may be a certain sequence of events you must follow to shut the unit off and turn it back on when working on or near the battery, or with the unit open. Also, be careful to neutralize any leaking battery electrolyte if necessary.
Having said that, you need to be careful, the diode and fuse are a good safety measure, but everything depends on the voltage that the UPS expects to be supplied by the battery. Not all of them work at a nominal 12 VDC. I've got an old SOLA unit that uses a 72-V battery to generate about 750 W of sinusoidal AC power; it weighs more than 75 lbs and is larger than a small microwave oven. One of my other units, a Best Fortress 600, has a LED display on the front that tells me, among other things, the battery voltage, and it reports a nominal 26 VDC when the unit is operating.
Depending on the make and model, some UPS units can sense their own internal temperature, and when that gets too high, they shut down. So even if you provide additional run time with external batteries, if the load is high enough, the UPS may not stay running long enough to make use of it. The better units have microprocessors inside that check operating parameters and rate of discharge, and they can decide when they've provided enough battery operation regardless of how much battery capacity you may have. Companies such as American Power Conversion, and probably all the other major UPS manufacturers, make units that utilize external power packs to extend their run time, but they are designed for this application.
If the required voltage is compatible with the external battery you want to use, the UPS may provide enough charging current to charge the external battery itself; it just may take longer to fully recover after a power failure. How dead is the internal battery? Is it just low on charge? Are any cells shorted or open? I presume the unit is not under any sort of warranty. Plug it in, turn it on, and let it charge without an external load for 24 hours, then measure the battery voltage and see if it has reached the proper level. If it has, you can probably connect your (fully charged and equal in voltage) external battery directly in parallel with the internal one. If not, remove the internal battery and try connecting the wires to the external one. Many use some sort of push-on spade lug connector.
Most car batteries are lead-acid and can emit hydrogen gas while being charged. This is usually not a problem in an automobile since it tends to sit outside most of its life instead of under your computer desk. UPS batteries are usually sealed and of the gel-cell type, in which the electrolyte is not easily spilled. They still have vents on top, but these are for emergency use only. They also are rated to operate near 120 VAC circuitry, which isn't usually present in the average car.
Another thing you should consider is the age of the unit. If the battery is dead, it's either been left off and uncharged for a long time, or it's past its useful life of five years. Go out and check on some new units of similar or higher capacity. You might find it's easier and cheaper to throw the old unit out and buy a newer, better one for less money than you might spend buying a battery and a charger, and connecting everything together. The modern units have hot-swappable, user-replaceable batteries, and some companies will even give you the option of returning the dead battery for proper disposal. APC even has a trade-up program that means it will accept your old UPS as a trade-in towards one of the new units. One thing's for sure, a new unit will definitely be lighter than any car battery!
There are some items to consider when purchasing or even using an UPS. Think about maximum capacity in watts and volt-amperes (VA); total run-time at full load; output waveform: square wave, sine wave or modified sine wave; switching time (AC to battery); recharge time; upgrade opportunity and replacement; control signals from the UPS to the computer system (this can be either a relay closure or full ASCII messaging).