The two spacecraft were 400 kilometres (245 miles) over central Asia near the Russian-Mongolian border when the shuttle commander, Robert Gibson, eased the mating devices for the Atlantis and Mir into first contact. "We have capture," said Mr Gibson, precisely on time at 1300 GMT. The Mir commander gave a similar message to his home base.
Television viewers around the world watched live. One screen visible to ground controllers showed the view of the approaching shuttle from the Mir. The other displayed what Mr Gibson was seeing - a greenish target approaching slowly. He had the demanding task of steering the 100-ton Atlantis to within 7.5 cms (3 ins) of the 123-ton Mir, at a closing rate no faster than 30 cms (1ft) in 10 seconds, while the two ships orbited the earth at 28,150 kph (17,500 mph). Cameras from Mir captured images of astronauts and cosmonauts waving and smiling from the Atlantis windows.
Mr Gibson had only two minutes to get the craft docked, so that Mir could maintain contact with a Russian ground station. The shuttle had no such constraints; it is in near-constant contact with Mission Control via a data relay satellite.
The danger was that the two craft might bump into each other with too much force, damaging the pressurised hulls.
The ships will orbit as a single unit for five days - 77 circuits of Earth - forming the largest man-made satellite ever.
When they come to separate, a mechanism will release the hooks securing the docking tunnel between the ships and springs will push them apart to a distance where the shuttle's jets can be used. The shuttle then will fly around the Mir to take photographs.
The Atlantis-Mir docking is the first of seven link-ups planned for the shuttle and Mir between now and the end of 1997, and it represents the first step towards the construction of the international space station by 2002.
Two Russian cosmonauts will transfer to Mir, as relief crew for Vladimir Dezhurov and Gennady Strekalov, who will return to Earth in Atlantis. Accompanying them will be the US doctor Norman Thagard, who has spent more than 100 days in space on board Mir, a record time in space for an American.
The Atlantis mission, therefore, will be a first in another respect: no other shuttle has returned to Earth with more crew members than it started with.
The symbolic rendezvous posed challenges unparalleled since 17 July 1975, when an Apollo capsule linked up with the Soviet Soyuz for two days of orbital detente during the Cold War. Because Apollo and Soyuz were smaller craft, that docking was technically easier, but political difficulties ensured that that moment of co-operation in space was to be a false dawn.
This time around, the hopes are higher. The ultimate goal is to spend the years 1997 to 2002 assembling a space station called Alpha in orbit, involving co-operation not only between the US and Russia but also including contributions from Canada, Europe and Japan. "We are on the verge of a historic moment in our lives," said Yuri Semenov, president of the Russian aerospace company RSC Energia.
Mr Semenov said that after the 1975 Apollo-Soyuz docking mission, there were plans to have the yet-to-fly space shuttle link with the yet-to-fly Mir, but politics interfered and the idea was scrapped.
However, the US and Russia may have trouble carrying out their ambitious plan in the face of relentless budget tightening by Congress and the Duma. Fiscal analysts at the US General Accounting Office last week set the price for space station Alpha at $98bn (pounds 62bn) - almost three times Nasa's current estimates. The Republicans in Congress have Nasa's current annual budget of $30bn firmly in their sights as part of their efforts to reduce government spending in th United States. Nasa needs this mission to be a resounding success if it is to prevent the space station from becoming budget cuts.