Bypass HttpContext MapPath method for Unit Testing

First, I know there are a ton of examples of getting around the HttpContext MapPath for unit testing on stack overflow already, but I still feel like my question is unique to what I've seen so far.

I already have an interface for my WebService

    public interface IWebService
    {
         Token GetTokenType(string url, string props, string values, string tokenType);
    }

My Production Implementation is:

    public class WebServiceImpl : IWebService
{
    WebService sts = new WebService("https://example.com", HttpContext.Current.Server.MapPath("~/Config/config.txt"));

    public WebServiceImpl()
    {
        sts.BasicAuthUsername = "username";
        sts.BasicAuthPassword = "password";
    }

    public Token GetTokenType(string url, string props, string values, string tokenType)
        {
            Token token = sts.GetTokenType(url, props, values, saml);
            return token;
        }
    }

And my Test Implementation is:

    public class TestWebServiceImpl : IWebService
    {
    public WebService sts;

    public WebServiceImpl()
    {
        //This is the line I need to somehow mock or stub or skip..
        sts = new WebService("https://example.com", HttpContext.Current.Server.MapPath("~/Config/config.txt"));
        sts.BasicAuthUsername = "username";
        sts.BasicAuthPassword = "password";
    }

    public Token GetTokenType(string url, string props, string values, string tokenType)
        {
            Token token = sts.GetTokenType(url, props, values, saml);
            return token;
        }
    }

The problems I've encountered with trying to use dependency injection on the HttpContext MapPath is that the WebService using it as a constructor will not accept any path I give it. Additionally, I can't see into this WebService class, but I believe it is going to try and make a call to the url even if I can get it to read the file. I also believe that it defeats the purpose of the unit test if I'm using it.

Therefore, my question is, can I mock the WebService within the interface (or some other way) so that I don't even have to use the MapPath and my WebService mock is good to go? Everything after this can be stubbed simply enough, it is just getting past this pesky MapPath.

A few things I've tried: Creating a mock object within the interface:

Mock<WebService> webService = new Mock<WebService>(bogus, constructor)

Stubbed the getTokenType() but still needed to instantiate the WebService.

Thanks in advance!