You'll probably see this kind of story being repeated across the press for the next couple of days.
This is because the ABCe have now changed access to their report so that normal people can't access them without registering (and being an employee of a member company).
The Daily Mail has somehow managed to add an extra 5 million readers in the last month. This is more than adding all the people who live in Birmingham. It's a bit unlikely to say the least.
Looking at the actual certificates, the Daily Mail's gone from 7.974m UK uniques last month to 8.316m UK uniques this month. That's an increase of 400k, which is 5%. That's good but definitely no the 19% that Brand Republic are reporting.
The increase has come from international traffic.
By UK traffic, the position is currently:
Guardian - 10.211m
Daily Mail 8.316m