The health campaigning charity wants to reduce illegal tobacco sales from its current estimate of 13% of the total market to 8% by 2010, and to 3% by 2015.
ASH director Deborah Arnott said tobacco smuggling was currently costing the UK economy about £2.4bn a year in lost revenue as well as undermining health policy and exacerbating health inequalities. Recent research commissioned by ASH found that one in four poorer people bought illicit tobacco products, compared with one in eight of the most affluent.
ASH is also calling for a more effective inland tobacco smuggling enforcement strategy, as well as stronger border controls. Greater collaboration was also needed between the NHS, local government, the police and UK Border Agency, it added.
The Association of Convenience Stores welcomed ASH's proposals.
Public affairs director Shane Brennan said: "The tobacco black market is one of the biggest risks to health in deprived communities. Tackling it has to be the centrepiece of the government's tobacco control strategy."