Copying cell data in one column when duplicate cells located in separate column

[Excelcious]

Hello all,

Had a look around and couldn't find a solution to this one. I've never used programs in excel before, only formulas, but I'm thinking I might need something more for this one.
I have two columns: Customer ID # and Customer Name.
The excel I have been given is combined from two different lists. I have duplicate contacts, (which need to be kept in) but the 1st list contained no customer ID numbers. When cells in column B contain the same name, I need the customer ID number to be copied in the blank cells in Column A. Sample attached.

I've got a few thousand lines so any suggestions would be most appreciated. Thanks!