The Information Commissioner's Office, which imposed the fine, said Better for the Country Ltd, one of the major Brexit campaign groups and better known under trading name Leave.EU, broke the law by not having the consent of the people to whom it sent text messages.
An ICO investigation discovered that Better for the Country had bought a list of phone numbers from a third party supplier but had failed to ensure that the supplier had obtained the data lawfully and had the necessary consent of those on the list.
ICO said those on the list should have had it clearly explained to them that they could receive promotional messages from a political campaign, but many had only consented to receiving messages about subjects such as leisure and home improvements.
Stephen Eckersley, ICO head of enforcement, said: "Political parties and campaign groups must follow the same rules as anyone else. That means they must have people’s permission before sending them text messages.
"Better for the Country did not have permission to send these messages. After considering all the options we decided that enforcement action was necessary."
In March, ICO fined David Lammy MP £5,000 for making more than 35,000 automated nuisance calls asking people to back his bid to be the Labour candidate for London Mayor.
And in December, Telegraph Media Group was fined £30,000 for sending hundreds of thousands of emails on the day of the general election urging readers to vote Conservative.Leave.
EU said it would appeal ICO’s ruling.A Leave.EU spokesman said: "There are a number of inaccuracies in the Information Commissioner’s assessment."
ICO said appeals must be lodged within 28 days of the fine being imposed.