First, why this isn't going to work:
A 1mm projectile has a cross-sectional area of less than 1/60th of a 7.62 NATO projectile. Assuming that your projectiles are cupronickel jacketed tungsten and about 10mm in length you're looking at a maximum of 0.15gm (2.3 grain) - about ~1/50th of the mass of the M80A1 projectile. And that means about 1/50th of the kinetic energy at comparable muzzle velocity too.
In order to increase the muzzle velocity to the point where the round will do a useful amount of damage you're going to need both a longer barrel, higher pressure or both. Yes the projectile will accelerate better, but the tiny CSA means that the same pressure will result in less force applied to accelerating the projectile. Longer barrel gives you more time for the acceleration, more pressure gives you a higher acceleration... and both require more material in the barrel.
And that's why weapons are the way that they are. We have some pretty awesome explosives we could use for propellants if we had stronger barrels... but the extra strength needed would make them heavier, so there's not much point.
Option 1: Better materials
To lighten up the weapon and reduce the amount of material required you'll need a much better material to work with than what we're using right now. There are some interesting options in high-impact ceramics for the bore lining but they tend to be quite heavy.
Alternatively you can reinforce the barrel with a layered composite material, high tensile but light weight. Wind some heat conducting threads through it to help with the bloom.
Option 2: Whole new tech
Two words: rail guns.
No, really. Man-portable electromagnetic rail cannons powered by nano-engineered graphene ultra-capacitors, firing 3mm flechettes at several times the speed of sound. The firing rails need to be fairly solid but the rest of the gun can be made of light-weight composites.
Option 3: Fun with alternative ammo
“What you need is something that doesn’t have much of a kick.” Goetz frowned. “Have you thought of trying them on splat guns?”
“Compressed-air guns that shoot little paint balls. Some of the guys in the department use ’em in a weekend war-game club they belong to.”
“Oh. Those things.” Phule shook his head. “I always thought they were more expensive toys than weapons.”
“Some of those ‘toys’ are fully automatic and have a muzzle velocity of over four hundred feet per second,” the chief informed him.
“Really?” The commander raised his eyebrows in surprise. “I didn’t know that. Still, I’m not sure what good it would do to hit someone with a paint ball in combat, no matter how fast it was going.”
“Well-” Goetz grinned wolfishly, easing himself back onto his bleacher seat “I just might be able to run down a source for some HE paint ball loads.”
Asprin, Robert. Phule's Company.
OK, so painball guns probably aren't your best option. But the idea of using alternative ammunition to greatly increase the effectiveness of an otherwise mediocre weapon bears some examination.
Look at some of the interesting rounds proposed for the AA-12 shotgun, most of which can be fired from any 12-gauge shotgun. The FRAG-12 fin-stabilized grenade is always the first to spring to mind, but they had a ton of interesting rounds including flechette packs and so on. And there's nothing stopping you from making more.
And yes, you can actually fire some pretty nasty things out of a paintball marker. Anything that you can fit in a .68 cal sphere actually. And hey, they're a technically less-lethal firearm, so that's a bonus.
...or just rethink your premise.
Honestly I'm having trouble coming up with a plausible series of events that would force us away from the current tech... except perhaps the invention of new materials that make even better firearms possible. When you look at the decades of cheap knockoffs of the AK firearms range it's clear that you don't need high tech, high energy-drain manufacturing processes to churn out usable firearms... albeit cheap, nasty ones with no quality control. Things would have to fail pretty seriously before we'd be forced to look for alternatives.
Incidentally, we have materials to make lighter weapons already, and modern rifles aren't generally all that heavy. A stock standard AR-15 weights in at 3kg (6.6lb) soaking wet... I mean fully loaded. A little more if you got that extended mag and scope option. And if you're determined you can shave that down quite a bit by swapping in some custom parts - lighter barrel, carbon fiber hand guard, lightweight BCG... you can get it down about half a kilo (a little over a pound) without much fuss.
Fun as this was, I fear I might have missed the point of your question. If so let me know and I'll rethink this.