Howard Hill shot an elephant with an arrow from a bow about 125# pull I believe and got close to 36" penetration. I would take on the t rex with a 250# crossbow drawing about 30". Not near as much energy as a rifle round but razor sharp penetration with many thousands of pounds per square inch force at the tip of the broadhead.
Bradley infantry vehicle has suitable weaponry. And sufficient armament that you would only need to hit it once and then take a nap while it dies. No need to waste ammo.
If the T-Rex were a thread on this board, we could just wait for someone to Google it and then give it a shot.
This is the gun I would use, known as the German 88. Good for tanks up to a couple thousand yards and aircraft up to 30,000 feet.
So, also good for any flying beasties that come at you. Best used on an open plain, so you can spot your target at a distance and load. With a good reload team you could fire 15 rounds a minute, assuring that you can take your limit. 
I’d just rely on my spear and magic helmet.
*“Be vewy vewy quiet. I’m hunting T-wex!” *
What if the T-Rex were on a treadmill? bap ow!
You all make me shake my head and laugh. You assume T-rex was like some type of bad-ass raptor, who closed in on hapless prey like a Peregrine Falcon on a high-speed dive.
Uh uh <shakes head slowly, and laughs>
I’m quite confident that T-rex were more like big chickens. As such, I just need a water pistol and a helmet-cam (chickens don’t like to be squirted with water and I need the helmet cam to video this funny “YouTube moment.”)
If I came across a flock of T-rex in the clearing ahead, preening and pecking the ground for seeds and berries, I’d prime my pistol and turn on my helmet cam. I’m not an idiot, I realize not just any squirt gun would suffice. I’d need a model that would have me do some serious squirting, like this baby!
I’d sneak up behind the big ol’ mega-chickens till I was within squirt range. Then I’d scream, “SHOO!!!” and commence blasting away. That’s when the hilarity ensues! They’d start falling over each other as they scampered away, chicken-stepping and hopping, tiny arms flapping akimbo, squawking “boc boc boc boc boc.” And I’m sure at least one of them would die from cardiac arrest, fulfilling the OP requirement. Yes, I’m quite certain that’s exactly how it would play out.
IIRC granted its a book but Crichton usually did decent research. In Jurrasic Park, Muldoon the game warden wanted these
for quick takedowns on big dinos.
There are at least two schools of thought on whether or not the Rex was:
- Double the elephant weight
- Capable of fast running or any running at all
Nevertheless, it is probably quite reasonable not to consider anything significantly lighter than the BIGGEST NEs and Magnums, even on mammals like hippos and buffaloes there is a very good chance of not returning home even with a nicely placed shot. Rex is different, not only penetration has to be good, incapacitating it would be a serious problem.
Whether You subscribe to the Hydraulic shock theory and / or primacy of penetration the arrows, with utmost respect, are lacking.
I would not try that.
Since this thread was poked with a stick…
What you do is hand the T Rex your gun, and then when he fires it, it knocks him on his ass, Then you run over it with an M1 Abrams.
Animals don’t like fire. Fire is a great deterent. Streaming fire into his face while he approaches will send him running away, not trying to bite down on the source of the fire. Does he even have enough brains to figure out there is a “source” and that if he eats you, you won’t be shooting him any more?
Depends on the circumstance. If you are going T Rex hunting, then you can plan your strategy and pick the high ground. But from 2 miles away, is the T Rex even going to know you are there, or care? That’s 2 effing miles away. There’s a tasty whateverasaur right over there. Even if you do shoot him, he won’t know where it came from or what happened.
The more important scenario is that you run into a T Rex on your prehistoric expedition that is close by and ready to try you out for a snack. You don’t have 2 miles, you have 100 yards, at most. That is the “what are you packing?” scenario that matters.
This would be my weapon of choice. 
Initially misread that as M47 Pattons, and was like “Man, that dude don’t mess around!”
Guns intended to kill pachyderms are typically of fairly large caliber. The bullets are designed NOT to expand. They need to pierce through the tough hide, and then a lot of muscle to get to something vital. Double barreled rifles are favored for a quick followup shot, as wounded pachyderms are prone to charging.
Serious Answer
Bolt action rifle. I believe you could start with the .375 Ruger and work your way as far down the list as you’d like.
If I had to go drop a t-rex, I would prefer the Patton myself…
Why two .30 and one .50 cal machine guns on the Patton? Why not three with interchangeable ammunition?
I think the two .30s were mainly for anti infantry work, while the .50 would be either antipersonnel or a poor man’s anti aircraft gun. Might just be because there was more room where they put the .50.
/Derail
How about a multiple fire grenade launcher? I’d think that would blow huge holes in it or, if nothing else, deafen/blind it.