If I'm reading your post correctly, you can't install  2007 on top of 2003.

I am trying to migrate to MOSS 2007. I thought I could do this on the
same server? Is this incorrect? I thought that I would get the dialog
box that asked me what type of upgrade I wanted to perform.

You cannot do an in-place upgrade of SharePoint.

What are you installing when you get the error? You say you are trying do
the upgrade on a single server, but you want to do a gradual upgrade. If you
do an in-place upgrade, which is upgrading 2003 to 2007 on a single box, it
will upgrade the entire install, not just parts of it. Are you installing
the binaries of 2007 when you get the error? Have you applied SP2 to your
current 2003 install  first?

Take a look at this TechNet article on conducting an in-place upgrade:

technet.microsoft.com/en-us/library/cc263212(TechNet.10).aspx

Pay special attention to the Before you install  section. You have to make sure
that all pre-reqs are installed and the SPS2003 web service is Off-line BEFORE
running setup. If you haven't done that I believe you get the kind of error you
describe.

I am trying to install  the install the binaries. I have SPS 2003 with
service pack 3 and WSS 2.0 with service pack 3 installed.

I am trying to do a gradual upgrade.

is the web service off line when you try to install  the
binaries?

.NET 3.0 and ASP.NET 2.0 need to be installed and activated first also.
Otherwise it assumes an upgrade isn't possible.

Yes both are installed...................

I have stopped the following:

* Microsoft SharePoint's Search

* SharePoint Portal Administration

* WWW Publishing Service

Tried to install  the binaries again and got the same error message.

I haven't messed with 2003 in a very long time, but stopping the web service
in SharePoint is not quite the same as stopping the Microsoft services from
the services mmc.

What do you mean by the web service in SharePoint? I thought that was
the microsoft services in MMC.