In integrated pipeline mode, the correct handler is the svc-Integrated-4.0 handler.
I found this out by trial and error and by reading through the source code of the Nuget Server on Codeplex.
This article on how ASP.NET processes your requests is also very helpful in understanding what is taking place under the covers.
Additionally, for this answer to be the correct one, you need to ensure that the application pool is in Integrated Pipeline mode and not Classic mode.
The relevant section of your Web.config that will need to change is listed below, specifically the list of verbs that are now supported.
<system.webServer>
<handlers>
<remove name="svc-Integrated-4.0" />
<add name="Elmah" path="elmah.axd" verb="POST,GET,HEAD" type="Elmah.ErrorLogPageFactory, Elmah" preCondition="integratedMode" />
<add name="svc-Integrated-4.0" path="*.svc" verb="GET,PUT,POST,DELETE,DEBUG" type="System.ServiceModel.Activation.ServiceHttpHandlerFactory, System.ServiceModel.Activation, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" resourceType="Unspecified" requireAccess="Script" preCondition="integratedMode,runtimeVersionv4.0" />
</handlers>
</system.webServer>