SSAS project deployment fails with error "Internal error: Invalid enumeration value. Please call customer support."

Recently I have seen this error popping up more frequently. It can occur when you modify a SSAS project in Visual Studio 2008 (or BIDS 2008) and try to deploy it against an existing SSAS 2008 R2 database. The full error message will usually also mention a "storagemode" element and a metadata error for instantiating a dimension object:

Internal error: Invalid enumeration value. Please call customer support. is not a valid value for this element.
An error occurred while parsing the 'StorageMode' element at line 1 , column xxxx under Load/ObjectDefinition/Dimension/StorageMode
Errors in the metadata manager. An error occurred when instantiating a metadata object from the file, '\?C:Program FilesMicrosoft SQL ServerMSAS10_50.MSSQLSERVEROLAPData.0.db.3.dim.xml'.