Dido was built by Yarrow of Glasgow. She was laid down as a Rothesay-class frigate to be called Hastings on 2 December 1959, but in 1960, it was decided to complete the ship as one of the new Leander class, with the new name Dido. The naming ceremony for Dido took place on 21 December 1961, but her launch was delayed until the next day because of fog. Dido was commissioned on 18 September 1963. Total construction cost was £4600000.

The ship was 372 feet (113.4 m) long overall and 360 feet (109.7 m) at the waterline, with a beam of 41 feet (12.5 m) and a maximum draught of 18 feet (5.5 m). Displacement was 2,380 long tons (2,420 t) standard and 2,860 long tons (2,910 t) full load. Two oil-fired boilers fed steam at 550 pounds per square inch (3,800 kPa) and 850 °F (454 °C) to a pair of double reduction geared steam turbines that in turn drove two propeller shafts, with the machinery rated at 30,000 shaft horsepower (22,000 kW), giving a speed of 28 knots (52 km/h; 32 mph).

A twin 4.5-inch (113 mm) Mark 6 gun mount was fitted forward. While the Leander-class was planned to be fitted with the Sea Cat surface-to-air missile, Dido was completed with two Bofors 40 mm anti-aircraft guns as a temporary substitute until Sea Cat could be fitted. A Limbo anti-submarine mortar was fitted aft to provide a short-range anti-submarine capability, while a hangar and helicopter deck allowed a single Westland Wasp helicopter to be operated, for longer range anti-submarine and anti-surface operations.

As built, Dido was fitted with a large Type 965 long range air search radar on the ship's mainmast, with a Type 993 short range air/surface target indicating radar and Type 974 navigation radar carried on the ship's foremast. An MRS3 fire control system was carried to direct the 4.5-inch guns. The ship had a sonar suite of Type 177 medium range search sonar, Type 162 bottom search and Type 170 attack sonar, together with a Type 199 variable depth sonar (VDS).