In 1959 Packer won the English Schools 100 yards title. Next year she competed internationally in the long jump. She attended Didcot Girls' Grammar School (now Didcot Girls' School).
In 1962, she reached the finals in the 200 metres at the European Championships and in the 80 metres hurdles at the Commonwealth Games; she was also part of the 4 × 110 yards relay team that won two medals at these competitions. In 1963 she focused on the 400 metres, and already by her fourth 400m race ran a world-level time of 53.6 seconds.
When she was selected for the 1964 British Olympic team Packer worked as a physical education teacher at Coombe County Girls' School, New Malden, Surrey. At the Olympics she shared a room with long jump gold medallist Mary Rand. Packer was hoping to win the 400 metres, but was beaten into second place by Betty Cuthbert of Australia, despite setting a new European record at 52.20 seconds. Disappointed, Packer planned to skip the 800m event and had a shopping trip instead, until her fiancé, Robbie Brightwell convinced her to compete. Before the Olympics, Packer only had five domestic 800m races; she had taken up a longer distance to improve her stamina, and earned the third British spot at the last minute.
In her heat and semi-final Packer finished fifth and third, running 2:12.6 and 2:06.0 respectively, being beaten by French runner Maryvonne Dupureur, clocking 2:04.5 and 2:04.1. She thus started the final the second slowest of the eight contestants, having raced at the distance only seven times before. Packer was sixth at 400 m, lying behind Dupureur, she was third at 100 m and took the lead in the final straight, using her sprinting speed to take the gold. She broke the world record with a time of 2:01.1 minutes. Commenting on her win, Packer said "Middle-distance running for women was still in its infancy and the 800 m had only been run in Rome four years earlier for the first time. I knew nothing about the event but being so naive was probably to my advantage; it meant I did not have any limitations in my head regarding what I should or could do. Ignorance proved to be bliss." Packer's winning performance is featured in Tokyo Olympiad the official documentary of the games directed by Kon Ichikawa.
After winning the gold medal, she announced her retirement at the age of 22 and so had one of the shortest athletics careers of any Olympic gold medallist. It would be another forty years before another British woman, Kelly Holmes, would win the 800 m, despite British men being successful at the distance.
Later in the same Games, Robbie Brightwell won a silver medal in the 4 × 400 m relay. They later married and had three sons, Gary, a 400 m runner like his mother, and Ian and David, the latter two becoming footballers with Manchester City. She and Robbie were each appointed MBEs in 1965. In 2011 Brightwell published a book detailing their careers: "Robbie Brightwell and his Golden Girl: The Posh and Becks of Yesteryear." Packer now lives in Congleton in Cheshire.
In 2009, Packer was inducted into the England Athletics Hall of Fame. Ann was coached by Denis Watts and was a member of Reading Athletic Club when she was selected for the British Olympic team.
In 1966 Packer appeared in an experiment for the BBC TV history programme, Chronicle to see how far geese could walk in a day. She was chosen because however far the geese went, she would still be with them at the end.
Athletic personal bests: 100y 10.9 (1963), 10.8w (1960); 100m 11.7w, 12.0 (1960), 200m 23.7 (1964), 400m 52.20 (1964), 800m 2:01.1 (1964), 80mh 11.4 (1960), HJ 1.60 (1959), LJ 5.92 (1960), Pen 4294 (old tables) (1963).