This story is in the Shiva Purana, and you are correct that Chalantarana was killed by Shiva. ('Chalantarana' is a variant spelling of the original name 'Jalandhara') There is a chapter devoted to the entire battle scene, although I've only provided the lines that describe the death itself. From Shiva Purana, Rudrasamhita, Chapter 24:
Śiva held in his hand the wheel Sudarśana which he had made with his toe and got ready to kill him.
Lord Śiva hurled the discus Sudarśana which resembled a crore suns and the fire of dissolution.
Blazing the heaven and the earth, the discus hit Jalandhara and severed his head with wide gaping eyes.
The body of the son of the ocean fell on the ground from the chariot making the earth resonant. The head too fell. There was a great hue and cry.
His body fell in two halves like the mountain of collyrium split by the thunderbolt and hurled in the ocean.
The whole universe was filled with his terrible blood O great sage, the entire earth became deformed.
His entire blood and flesh, at the bidding of Śiva was taken to the hell Mahāraurava and became a big pit of blood there.
In case you are interested, here are the same lines from the original Sanskrit version:
sudarśanākhyaṃ yaccakraṃ padāṃguṣṭhavinirmitam |
jagrāha tatkare rudrastena haṃtuṃ samudyataḥ || 45 ||
sudarśanākhyaṃ taccakraṃ cikṣepa bhagavānharaḥ |
koṭisūryapratīkāśaṃ pralayānalasannibham || 46 ||
pradahadrodasī vegāttadāsādya jalaṃdharam |
jahāra tacchiro vegānmahadāyatalocanam || 47 ||
rathātkāyaḥ papātorvyāṃ nādayanvasudhātalam |
śiraścāpyabdhiputrasya hāhākāro mahānabhūt || 48 ||
dvidhā papāta taddeho hyaṃjanādririvācalaḥ |
kuliśena yathā vārāṃnidhau girivaro dvidhā || 49 ||
tasya raudreṇa raktena sampūrṇamabhavajjagat |
tatassamastā pṛthivī vikṛtābhūnmunīśvara || 50 ||
tadraktamakhilaṃ rudraniyogānmāṃsameva ca |
mahārauravamāsādya raktakuṃḍamabhūdiha || 51 ||