changing a cell value when the date changes to a new month

hi I've been stumped on this one for a while I am trying to set up an account reading system for water meters and it has a beginning amount (A1) and
end amount (B1). what I would like for it to do is when the new month comes around and I change the date on the spread sheet A1 will take the value of B1, but every time I try it with IF or Switch functions once I start to put a new amount in B1 it also changes A1 and I don't want it to do that.

Is there any way to accomplish this.