On March 24, the petition for certiorari was denied in the Strine v. Delaware Coalition For Open Government, Inc. case, ending the Delaware Court of Chancery's experiment with arbitration by their sitting judges.  (H/T Brian Quinn). 

As far as I know, however, sitting judges on the Delaware Court of Chancery still conduct mediation.  A Chancellor or Vice Chancellor does not mediate his own cases, but rather mediates the cases assigned to one of the other four judges on the court (if the parties agree to submit to mediation). 

More information about the Delaware Court of Chancery's mediation process is here.  The benefits of the mediation include:

• Expertise.  You would be hard pressed to find someone more knowledgable about Delaware corporate law and the merits of a Delaware Court of Chancery case than a sitting Delaware Chancellor or Vice Chancellor. 
• Relatively Inexpensive.  The fee is only $5,000 a day, for cases that are already on the Chancery docket, which is a decent amount of money, but is dwafted by the legal fees spent in almost all of these cases.  For mediation only cases (cases not already on the docket), there is a $10,000 initial fee and a $5,000 for each additional day.  
• Confidential.  All mediation proceedings are strictly confidential.

These are many of the same main benefits as the Delaware Court of Chancery arbitration, but, of course, in mediation, the judge is not making a decision, but rather assisting the parties in reaching a voluntary settlement.  

According to Steven Davidoff, in the Strine case, "the federal court found that the arbitration proceedings were effectively a civil trial, with no difference in judges, place or proceeding except the secrecy and the arbitral nature." 

Mediation, however, is quite a bit different than a civil trial.  While the comments of a sitting Chancellor may carry a lot weight with the parties, a mediator does not come to a determination for the party and the parties are able to walk away from the mediation at any time.

In short, judicial mediation carries many of the benefits of judicial arbitration, but the practice of judicial mediation seems to be more difficult to challenge.