So you've met Pure Hive. A pleasant bunch.

Either you talked them down into not attacking you temporarily, or you didn't, and now they are at war with you. Sure they are a guilt-free source of titanium and other elements, but it doesn't have to be this way, and he game gives you a way to make peace with them.

Explore their space. If you've talked them down from attacking you, you can explore their space in peace for a couple minutes, though they'll be groggy about you being there, but they are not that much of a difficult enemy even if they do.
Starting from the system with the debris field around the wormhole, follow the flinger to the sourth-east, and you'll end up in a system which is usually free from the Pure Hive. The flinger to the east will bring you to the system where you've most likely encountered them for the first time. Your scanners should pick out a system to the east from there, which by all means look like their home system, what's with the glowy-green planet and a bunch of Hive ships and stations there. The system has a flinger going quite ways to the north-east. Follow it and you should end up in another system.

The system is devoid of Pure Hive for some reason, but more importantly, you should pick out one or two more systems on your scanner south-east from there. Explore them, and you will discover a couple curiosities:

• There's a planet that is abnormally hospitable, or so says you scanner anyway. But it does not give you any resources or research, so it is completely useless to you, even if your biologist thinks you'd be able to settle there. But anyway, ignoring this completely irrelevant bit of info, we also discover
  
• An once-inhabited planet that underwent some sort of a solar-related catastrophe, and so they build a launchpad and a colony ship (or two) to escape to the nearest habitable planet which they could pick out with their primitive telescopes. Your astroscience officer will also tell you that, whatever their destination was, they missed it by mere fractions of a percent, so they probably flew some distance past their destination since then.

Before you continue reading below, please try and figure out the solution yourself.

Okay, so there was a colony ship that was launched from the ruined planet in the direction of some sort of a habitable planet, but it most likely overshot. Draw a line from the ruined planet to the hospitable planet, and then follow this line way into the void. Watch your radar, and you should stumble upon the colony ship. It is approximately halfway between the hospitable and the Abyssar station.

From here on now, it should be straightforward. Use 20 titanium to repair their ship. Watch them go in the direction of the habitable planet. Slowly. It takes maybe 5 minutes for them to reach their destination? Also try not to block their way lest you want to spend even longer than that.
Once they are orbiting the planet, talk to them, and they'll offer you a way to make peace with Pure Hive. Take their offer, and go back to the hivers.

Hail any of their ships [X], click on the dialog options as they appear, and they will agree to let you be in their space, and if you hail them again, they'll tell you that they will take some time to investigate your claims.

As the colony ship you've saved is still building the colony, go back to Celaeno to pass time, and then come back to the colony, you can do so immediately even. The colony is now built, and the colonists will award you a bunch of intermediate resources.
Go back to the hivers, and they will give you a bit of gold and their technology, unless you've farmed it out of them already, and they will also grant you the honor of being marginally less hostile to you from now on.