Remove Ranges Bounded By Blank Rows

Once a week I get a spreadsheet with 39 separate batches of data, only across 5 columns, and each batch is separated by a blank row. The data batches are not the same 'height' in other words some may comprise just 20 rows and others as many as 550 rows - also, each week they are different to the previous week.

I've little experience with Macros, and failed dismally in trying to do the following:

1/. Delete the rows containing the first batch of data - including the 'blank' row at the foot of that batch.
2/. Save the file in it's original name after deleting the rows.

Then I perform another task with the remaining data, before repeating the steps above

It sounded simple before I started, but my recorded macro (cringe) seemed to make the assumption that every time I wanted to delete the 'top' batch of data rows, it would be the same length or number of rows as the batch I had used to 'record' the macro - in recording the macro I used shift+end+down to identify the rows I wanted removed, but the macro seemed to think I wanted to delete (say) Row 8 to Row 278 - which was true of the 'height' of the first batch of data but none after that.