Founder Laurie Flynn said the milestone was surpassed on Friday afternoon, with membership numbers now at 4,100.
According to the party, its membership figure gives it more supporters than the Scottish Liberal Democrats
The assertion is based on a listing of the party’s membership on its Wikipedia page, which shows numbers at 4,085 as of December 2018.
Mr Flynn said: “Little did I think when I founded the party the impact that it would have.
“After exactly one week since our public launch, our membership has surged past that of the Liberal Democrats, a party which has been in existence for 150 years.”
The Liberal Democrats were formed in the 1980s following a partnership between the Liberal and Social Democratic parties.
Mr Flynn said Alba was “Scotland’s new political force”.
However, a Survation poll released on Thursday put the party at just 3 per cent support with the more than 1,000 respondents.
The survey done for newspaper publisher DC Thomson also found more than half of those asked believed Mr Salmond was detrimental to the cause of independence.
The SNP retained its runaway lead in the poll, which spoke to more than 1,000 people, with 49 per cent in constituencies and 38 per cent on the regional list.
On Thursday, the former first minister said his party’s membership figures were one of the reasons why it should be permitted into the leaders’ debates, and refused to rule out taking legal action if Alba was not invited.