ASP.NET Web API - 404 On Delete

I am trying to implement a Delete method on a Web API controller. However, I always get a 404 - Not Found. At this point, I have GET, POST and PUT methods that are working just fine. I've been reading a handful of the other SO posts about the same issue - just none of them are working.

The Controller Action

public virtual HttpResponseMessage Delete(string customerId)
{
    adapter.RemoveCustomer(customerId);
    return Request.CreateResponse(HttpStatusCode.OK, "The customer was deleted.");
}

The AJAX Request

function remove(customer, success, error) {
    var url = '/api/Customer';
    var data = JSON.stringify({ 'customerId': customer.CustomerId });
    $.ajax({
        url: url,
        type: 'DELETE',
        data: data,
        contentType: 'application/json'
    })
    .done(function (data, textStatus, handler) {
        success(data);
    })
    .fail(function (handler, textStatus, errorThrown) {
        error(errorThrown);
    });
};

The Web.Config

This is my web.config file. Except for the modules section, everything is the same as when I created the project:

<system.webServer>
    <validation validateIntegratedModeConfiguration="false" />
    <!--<modules runAllManagedModulesForAllRequests="true"></modules>-->
    <modules>
      <remove name="UrlRoutingModule-4.0" />
      <add name="UrlRoutingModule-4.0" type="System.Web.Routing.UrlRoutingModule" preCondition="" />
    </modules>
    <handlers>
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" />
      <remove name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" />
      <remove name="ExtensionlessUrlHandler-Integrated-4.0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_32bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness32" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-ISAPI-4.0_64bit" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" modules="IsapiModule" scriptProcessor="%windir%\Microsoft.NET\Framework64\v4.0.30319\aspnet_isapi.dll" preCondition="classicMode,runtimeVersionv4.0,bitness64" responseBufferLimit="0" />
      <add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="GET,HEAD,POST,DEBUG,PUT,DELETE,PATCH,OPTIONS" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
    </handlers>
</system.webServer>

I am using IIS Express, but the issue still occurs if I switch back to Visual Studio Development Server.

The Raw HTTP

Here is the raw HTTP request captured by Fiddler:

DELETE http://localhost:63654/TestMvcApplication/api/Customer HTTP/1.1
Host: localhost:63654
Connection: keep-alive
Content-Length: 49
Accept: application/json, text/javascript, */*; q=0.01
Origin: http://localhost:63654
X-Requested-With: XMLHttpRequest
User-Agent: Mozilla/5.0 (Windows NT 6.1; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/27.0.1453.116 Safari/537.36
Content-Type: application/json
Referer: http://localhost:63654/TestMvcApplication/
Accept-Encoding: gzip,deflate,sdch
Accept-Language: en-US,en;q=0.8

{"customerId":"e107e2dc20834545ae209849bff195f0"}

And here is the response:

HTTP/1.1 404 Not Found
Cache-Control: no-cache
Pragma: no-cache
Content-Type: application/json; charset=utf-8
Expires: -1
Server: Microsoft-IIS/8.0
X-AspNet-Version: 4.0.30319
X-SourceFiles: =?UTF-8?B?QzpcVXNlcnNcdHBhcmtzXERvY3VtZW50c1xHaXRIdWJcVGVzdE12Y0FwcGxpY2F0aW9uXFRlc3RNdmNBcHBsaWNhdGlvblxhcGlcQ3VzdG9tZXI=?=
X-Powered-By: ASP.NET
Date: Tue, 02 Jul 2013 13:11:52 GMT
Content-Length: 220

{"Message":"No HTTP resource was found that matches the request URI 'http://localhost:63654/TestMvcApplication/api/Customer'.","MessageDetail":"No action was found on the controller 'Customer' that matches the request."}

This is an open source project for teaching myself. I have checked in the latest in case anyone wants to see the complete source.