The private key always includes the public key.

What you might really want is Signing. Using the same .NET classes, you can sign data with your private key and verify the signature on the other party's side with the public key (which obviously doesn't contain the private key).

    public static string Sign(string data, string privateAndPublicKey)
    {
        byte[] dataBytes = Encoding.UTF8.GetBytes(data);
        RSACryptoServiceProvider provider = CreateProviderFromKey(privateAndPublicKey);
        byte[] signatureBytes = provider.SignData(dataBytes, "SHA1");
        return Convert.ToBase64String(signatureBytes);
    }

    public static bool Verify(string data, string signature, string publicKey)
    {
        byte[] dataBytes = Encoding.UTF8.GetBytes(data);
        byte[] signatureBytes = Convert.FromBase64String(signature);
        RSACryptoServiceProvider provider = CreateProviderFromKey(publicKey);
        return provider.VerifyData(dataBytes, "SHA1", signatureBytes);
    }

    private static RSACryptoServiceProvider CreateProviderFromKey(string key)
    {
        RSACryptoServiceProvider provider = new RSACryptoServiceProvider();
        provider.FromXmlString(key);
        return provider;
    }