ADO RecordSet Save generates "Access Denied" error (80070005)

I have an Access DB with some code that has been (and still is) running fine
for a long time. I have recently set up a few Windows 7 x64 machines, and
this bombs on those machines only. (other machines are Vista 32-bit).

The code is using an ADODB RecordSet to create a new record (AddNew), sets
the various columns to appropriate values, and then does a Save. I get the
Access Denied error on the Save command, but when I examine the table, the
record has been created successfully.

Any ideas on what is different between the two environments that might be
causing this?

Any techniques I can use to further debug?