<p>We have some really old code that calls WebServices using behaviours (webservice.htc), and we are having some strange problems... since they've been deprecated a long time ago, I want to change the call.</p>

<p>What's the correct way of doing it? It's ASP.NET 1.1</p>

While I'm not 100% sure what the Web Service behavior does, I recall it allows client-side script to call Web Services, which would make AJAX it's contemporary replacement.
Since you're using .NET 1.1 how about using
Ajax.NET Professional
to connect to the web services?