Permissions problem accessing SQL Server from Web Service on Windows 2003

[RichardF]

I have a Web Service that accesses a SQL Server database.

IIS, SQL Server and the Web Service are all running on the same
machine.

As I have done on other machines, I gave the appropriate permissions
to the IIS_WPG user group.

However, on this machine, when my web service tries to open a
connection to the database I get the following message...

..Net SqlClient Data Provider: System.Data.SqlClient.SqlException:
Login failed for user <DOMAINNAME>\<MACHINENAME>$'. at
System.Data.SqlClient.ConnectionPool.GetConnection(Boolean&
isInTransaction) at
System.Data.SqlClient.SqlConnectionPoolManager.GetPooledConnection(SqlConnectionString
options, Boolean& isInTransaction) at
System.Data.SqlClient.SqlConnection.Open() at
MyWebService.OpenTheDBConnection()

Where <DOMAINNAME> and <MACHINENAME> are the domain name and machine
name of the machine where I am running everything.

Any ideas why it is using <DOMAINNAME>\<MACHINENAME>$ instead of the
IIS_WPG group?

Thanks for any help.

RichardF