The beginning of the novel, set in 1999 (2019 in the second edition), finds Nigel Walmsley, a British scientist and astronaut for NASA, sent to attach a thermonuclear bomb to an asteroid or comet named Icarus which is on a direct collision course for India - only if it is a rocky asteroid and not a slush ice core style comet.
Icarus turns out to be large, solid, and made of a nickel-iron composite. Nigel is instructed to plant the 50 megaton weapon and leave (so it can be detonated). He persuades Mission Control to let him put it in a large fissure he discovered, so it would be even more effective. They let him.
In the fissure, Nigel discovers strips of metal worked in obviously artificial patterns. Awestruck at this evidence of extraterrestrial intelligent life, Nigel begins exploring. Icarus is made up of a number of hollow shells, making the asteroid's mass far less than predicted. Presumably this makes Icarus less dangerous, allowing Nigel to spend time exploring this historic artifact. However, NASA claims that the demolition has to go forward, that Icarus would somehow skip off the atmosphere and land in the Indian Ocean and cause even more damage through the resultant tsunami. This of course is an obvious lie, and Nigel convinces his partner of that. They hide the nuke and spend the next week retrieving artifacts and materials before they finally set the nuclear bomb off and turn Icarus into rubble.
15 years after their discovery the Icarus artifacts have yielded little, and Nigel's delayed detonation of Icarus has distanced him from NASA and other people. Nigel's partner, Alexandria, has developed systemic lupus erythematosus, an oft-fatal disease caused by pollution. An anomaly over by Jupiter distracts Nigel: something, nicknamed 'the Snark', is repeating radio broadcasts. Alexandria is distracted by the mechanics of selling American Airlines to some Brazilians.
The anomaly fires its fusion engines and reveals itself to the satellites around Jupiter. As a probe vessel, the directing computer could not afford to ignore the satellites' radio emissions before it moved on to Earth.
Eventually the JPL team locates it around Venus. Nigel arranges to hijack the communications, transmitting his own signal (a binary sequence of prime numbers relating the Snark's trajectory). The Snark receives the signal as a sign of non-hostile intentions and transmits back. It also reaches out through Nigel's medical implants to his dead partner's more elaborate ones, and commandeers her body to explore and learn about Earth. Thus the initial tentative transmissions blossom into a largely one-way torrent of information for the Snark. One day, it asks to visit Earth. A compromise is worked out: the Snark will orbit the Moon until trust is built up. As Nigel is already fully informed, and everything about the Snark is being kept a state secret, he is assigned to pilot the space ship meeting the Snark - which will be armed with another nuclear weapon.
Nigel meets the Snark; he is under orders to attack it. The Snark disables the chemical weapons and begins talking to Nigel. It says that organic civilizations and species are inherently unstable; they flash brilliantly and commit suicide sooner or later. The autonomous machines they craft live on long after them, going on and evolving. But they cannot truly compete with the organics, who live "in the universe of essences". That is the reason for the Great Silence.
Nigel's superiors order him to use the nuclear weapon. He refuses. They override him and fire it anyway, knowing that its detonation will inevitably kill Nigel. If it hit the Snark, it would be badly disabled, so it flees the Solar System faster than the missile can follow.
The decision to fire is covered up, and the version of the Snark's visit fed to the public is markedly different from what Nigel actually experienced. Nigel blackmails NASA into letting him go to the projects on the Moon; the Snark had directed a transmission at Mare Marginis for unknown parties, and Nigel wanted to find those parties.
Four years later, in 2018 (2038), Nigel is now based on the Moon. A fellow astronaut, Nikka, is involved in a crash that accidentally discovers a still active alien spacecraft wreck in the Moon's Mare Marginis - a spacecraft suspiciously armed with a once-powerful anti-spacecraft weapon. Nigel and Nikka become lovers during the course of exploring the wreck, which proves to have a functioning computer with a direct neural interface. Nigel, and several others, experiment with the computer's neural hook-up, and leave fundamentally changed by it - the computer becomes inert and unable to reveal any more about its creators. Meanwhile, on Earth, some surprising experiments in human genetics conducted by the aliens are discovered alive in North America.