But Stephen Timms told a special debate in the Commons on the issue that now "we are making rapid progress''.
Timms confessed that in 1987, when working in the IT industry, he had written an article predicting there would be 600,000 broadband connections in the UK by 2000.
"In fact it took longer than that: 600,000 was reached by the time of my appointment to my present position a year ago this month," he admitted.
"From 1987 it took until last October to reach one million broadband connections, but it will have taken only nine months to add the second million, which I expect us to achieve in the course of this month.
"Today there are more than 1.9 million connections - one million via cable modems and 900,000 via ADSL - and the number is increasing by well over 30,000 a week, which is one of the fastest rates of growth anywhere."
Independent research has identified the UK as having the second-largest broadband network after Germany, said Timms.
The minister said more than 70% of households can access one of the mass-market broadband services, with BT planning to take ADSL coverage to 90% of UK households shortly.
However, he admitted there were problems of access in some rural areas.
While satellite broadband was available, it was expensive so the government was trying to aggregate public sector demand to make it commercially viable for commercial providers to extend broadband services to rural communities.
Timms' shadow minister, Andrew Robathan, admitted: "The situation is improving, so I do not knock the government.''
Liberal Democrat spokesman Vincent Cable expressed concern about the growing division between young people who used IT and broadband and older people who did not.