DNS cannot resolve to individual pages; however, your server can decide what is served up to a client who does not request a specific page. As you mentioned, this is usually index.xxx or default.xxx, depending on what server you use. But, there is a way to do what you're looking for.
Step 1: DNS
So, first things first; as Matthew Brookes mentioned in his answer, you'll need to create an A record for that IP address. If this is the only thing on that domain, you probably want to use @ as your host, which is a wildcard. If I visit example.com, or thisrecorddoesnotexist.example.com, it will direct you to the IP you set in this record.
This will get you the example.com/script.aspx address, but won't take people directly to it, which brings us to Step 2.
Step 2: Default Document
I'm assuming you're using IIS, because of the aspx filename. Anyways, you can set the Default Document of your web site to something other than default.aspx or default.htm.
In IIS properties, under the web site in question, click the 'Documents' tab (Note: This is in IIS6, don't know if 7 is different) and add your script name to the default documents, at the top. That way, whenever someone visits example.com, they will be presented with your script by default.
Step 3: Couldn't do Step 2
Not everybody has access to configure their server, especially if you're using shared hosting of some kind. So, as a backup, you could always create a default.aspx document that redirects the user to the correct page. This wouldn't require any additional server configuration.
Step 4: Profit
So, I think that should do what you're looking for.