You can certainly reuse the URL. Simply point the name servers to your new IIS server. You can even intercept requests for .PHP URLS and redirect them to corresponding .ASPX URLs - though this should be a temporary/migration solution; you don't really want to stick with PHP URLs for the longer term if you are actually using IIS/.NET. Are you only interested in keeping the domain or are specific URLs important?
It would be a good idea to redirect (HTTP 301) some or all requests to the original (PHP) pages to the new pages.
However, .NET is not necessarily any more secure than PHP anyway. It depends on the developers implementation; either platform can be made equally secure. Clearly, if the current site is badly designed, and you have been chosen to write a replacement and .NET is your preferred platform, then go ahead - just make sure you don't repeat the same mistakes in .NET.
What do you mean by secure anyway? Is the site an secured application or portal. That is, is it the security of the application you are concerned about, or the security of the underlying web server (apache)? Again the same argument applies - both Apache and IIS can be made equally secure if configured correctly.