I read this story in a science fiction anthology. I think I found it in a public library somewhere around the late 1990s. I don't remember the guiding theme of the collection (if there was one), nor anything about the title and cover art. I'm pretty sure it was not a single-author story collection, for what that's worth. I don't recall what other stories were in there, but I strongly suspect that the one story I'm asking about was not by one of the "Big Name Authors" of the 20th Century, such as Poul Anderson or Isaac Asimov, or else I probably would have run across it again, since then, in a collection of one author's shorter works.
I do remember the plot pretty well. Here are the points that stick in my mind:
The main character is an AI. I'm not sure any other character ever gets a single line of dialogue -- but if they did, it would only be in a flashback set before the main action of the story. The AI had a name, but I can't recall what it was. Basically, this AI is the brain of a much-slower-than-light starship which is making the long, long haul from Earth to some other solar system. (I don't remember which star, but probably one of the G-class stars that resemble our own sun and are located not-too-far-away from here. Tau Ceti springs to mind as an obvious example, but it may not have been mentioned in the story.)
The backstory went this way: Something terrible was threatening to wipe out everybody on Earth. (Implicitly, this had been in what we would call "the near future," as opposed to thousands of years from now.) I think there may have been a realization, many years in advance, along the lines of "our sun is going to go nova fairly soon," but I could be wrong about the details. At any rate, it was the sort of catastrophe which human scientists could see coming, way ahead of time, but could not possibly prevent.
Humans had never gotten outside our own solar system. (Heck, I'm not sure if they'd ever even made it as far as "walking on the surface of Mars.") So whatever was coming was almost certain to make Homo Sapiens go suddenly extinct . . . unless extreme measures were taken to relocate a bit of the human race to somewhere very far away.
A decision was made to spend money like water on an interstellar colonization project. However, since nobody had ever found a way to cheat the laws of physics with faster-than-light capability, nor even a workable way to accelerate a ship up to a significant percentage of lightspeed, this colonization ship would probably take tens of thousands of years to get where it needed to go. (Even assuming the first target system actually turned out to be a good place to colonize, which was by no means a given, and then the ship might need to keep coasting through space toward a secondary target.) The figure that sticks in my head is "a hundred-thousand-year voyage," but I could be wrong about that.
For most of the story, it is not clear to us exactly what cargo the spaceship is carrying. We only know that the AI is programmed to be obsessive about completing the assigned mission, no matter what. We also can tell that it doesn't have anyone to talk to during the long trip -- in other words, this was not a classic "generation ship" with a human crew running around on it and dreaming of the day when their posterity will walk on a new world.
The AI finally enters the target system and has to decide if it has found a suitable place to land. There are two planets in the habitable zone, either of which might be a decent fit for the parameters it's supposed to measure them against. Somewhere along the line, however, the AI has been shocked to realize it has forgotten exactly what its all-important mission is meant to achieve! (The writer makes it clear that this is not the AI's fault -- the result of millennia of exposure to cosmic radiation, etc., has gradually eroded whatever storage medium was used for its memory.) But there's enough redundancy that the AI still remembers that landing on an Earthlike planet is a necessity, and so it finally picks the candidate which seems to have a higher percentage of water on the surface. The AI is fairly sure that lots of water is desirable, even if it can't remember why that matters!
The ship comes down on dry land, in an area which I believe is described as having lots of green vegetation. Then the AI starts defrosting an internal storage area which has been on ice for the entire trip, and the author (speaking as an omniscient narrator who knows things the AI doesn't) assures us that this is where the careful planning of the ship's engineering design team really manifests itself. Those engineers didn't have any way of knowing exactly what sort of damage the ship and its memory banks would suffer in traversing the light-years, but they could reasonably anticipate that there might be some serious wear and tear. So they hid a backup copy of all the key instructions for "what to do when you get there" inside the sealed-shut, frozen storage area, where it would be as protected as possible from outside influences, and even the AI could not access that data until the ice started to melt (or something along those lines).
In the final paragraph, the AI reflects that, according to this new data it is now studying, the next major stage in the project will begin about nine months from now. (Implication: It has some frozen human embryos along, and once they're warmed up, they will start gestating inside the artificial equivalent of a womb until they are ready to be "born.")
As I said, this was pretty much a one-character story, with the character being the Artificial Intelligence that flies the ship for thousands (or tens of thousands) of years. Does anyone recognize this one?