Jump started car. Battery was dead .jumper cables

Did you replace the battery as well? If both cables are badly melted it is likely the battery has suffered damage as well. Do you have access to a Multimeter? Set it to Volts DC. With the engine OFF put the red pin to the red positive battery terminal and the black pin to the black negative terminal. If the battery is good you should get a reading of around 12.6 volts or higher. Anything below 9 volts and the battery is probably worn out or damaged.
You will also need to find out what caused the cables to melt. There is most likely a 'Resistance' issue somewhere in the electrics. This is where the voltage is restricted and cannot flow correctly. This generates heat and is possibly the reason the cables melted. Have you 'Jump Started' the vehicle recently? Connecting jumper cables the wrong way around can have a similar effect.

Hi. it sounds like you may have a faulty alternator get it tested at a local garage. Match your + and - on the battery terminals .

I have never seen a fusible link on a Volvo.

Use jumper cables from another car-put the positive jumper cable on the post on the starter where the battery cable connects, then the negative jumper cable to a ground on the frame of your car. Turn on the car you are jumping with and let it run about 10 minutes. It might charge up your battery in the trunk enough to open the trunk.

if it is like my Holden omega which has the battery in the boot there will be a jumper post in the engine bay that you jump to a car like when jump starting cars .
Or put a charged car battery jumpered to jumper posts with a jumper cable .
Check engine bay for jumper posts

Check battery cables are correct ! Prime suspect here ! Alternator may be damaged and shorting .. Don't drive any car that will melt jumper cables ! Something very basic wrong here.

Your jumper cables are most likely red and black. Locate the positive and negative terminals on your battery, they are marked ( + ) or pos.,that's the red one, and ( - ) or neg. that's the black one. Conect one end of your cables to your battery (doesn't matter which one you do first) make sure you are connecting the other end to the same terminal on the dead battery (that is to say positive to possitive) now do the same with the other cable. There may be a little bit of a spark that's normal when you make the final connection. The danger is in taking them off, do it in the same order that you put them on, if you take both cables off at the dead battery and they touch you are going to be having a bad hair day. the thing that most people don't get is you have to have a good clean convention. If you don't try wiggling the clamp on both batteries to make them dig into the terminals. Also there is a huge difference between someone who left the lights on in parking lot and someone with a dead battery on the side of the interstate. The guy in the parking lot has a dead battery, the guy on the side of the road has a dead battery and a dead alternator, he needs a tow truck, not a jump.

THE JUMPER CABLES WERE OF A TOO LIGHT OF GUAGE WIRE. THERE SHOULD BE A BATTERY FUSE LOCATED IN THE FUSE BOX IN THE DRIVERS SIDE ENGINE COMPARTMENT.

with a new starter, this sounds like a bad battery cable or terminal end. the easier way to figure this out w/out checking the voltage drop across a whole battery cable is to affix the jumper cables strategically to find the bad one. Put jumper (+) on batt (+) and jumper (-) directly onto a piece of unpainted engine metal and not in the way of moving parts. If it starts this way then you've got a bad ground cable. If it didn't start, put jumper (+) on the hot post of the starter solenoid (there are 2 posts, only touch the one w/ the fatter cable going to it) or on a remote jump point if equipped, and put jumper (-) to batt (-). If it now starts you need a positive battery cable, or just a battery-to-starter cable (if separate)