Offbeat puzzle #1

At a Manhattan (N.Y. USA) chess club, White, a wealthy businessman, was playing Black, a noted chess master, for the usual stakes of $25 per game and the usual handicap of a Rook and move.  Black, who could have crushed his opponent at will even with the handicap, allowed White to win about one out of five games and kept the others close.  This kept White happy, and he cheerfully gave up about $100 per night, twice a week, for the privilege of thinking that he was playing the master close at the odds (when in fact, he would have been lucky to win one game in 100 if Black had played hard).

In the given game, Black carefully manipulated the position to where he would win by one tempo. At the move before the position listed below, Black excused himself and went to the washroom.  As White studied the position, he got a clever idea.  

While White waited, a couple of players at the next table were discussing a recent game.  White listened and watched as they talked and moved pieces around.

Fred: So he played g6 on move three of the Ruy Lopez!

John: 1 e4, e5, 2 Nf3 Nc6, 3 Bb5 g6? 

Fred: Yep.  

John: How did you handle it?

Fred: I played 4 d4, and he played exd5.  Then I played 5 Bg5, he played f6, I played 6 Bh4, he played g5, I played 7 Bg3, he played h5, I played 8 h3, and about that time, he realized that he wasn't in a good position.

John: I'll bet.  How long did it take you to win?

Fred: About 30 more moves.  But I usually beat this person anyway.

As the players at the next table continued their discussion, Black returned.  White made played Kf1, arriving at this position. 

White held up a hand before Black could move  the pawn, and he spoke: "I'll pay you five dollars if you let me give you a piece before you make your move."

Black was naturally suspicious.  "You'll give me a piece, a black piece?"

"Yes," White replied.

Black frowned.  "You could place a Queen or Rook at g1.  Then it would be your move, and you'd take it and draw.

"The piece I give you won't put me in check,"  White said.

"And it will still be my move?" Black queried. 

White nodded.  "It will still be your move."

"And it will be a legal position?" Black insisted.

"It will be a legal position", White affirmed.

Black looked for a minute and figured out what White had in mind.  Already being up $75 for the night, he decided that it was better for "business" to let White have his fun.  "Okay," he said.  "I'll take five dollars and a free piece."

White placed Black's new piece on the board, and Black, after studying a while, made a big fuss about being hoodwinked (as he could no longer win), much to White's delight.  White lost another game thereafter and cheerfully paid $105 for the evening's fun.

Here's the diagram again:

What piece did White give Black, and where was it placed?

Please do not post the solution in the replies. 

If you can't figure it out, the solution is White's 6th move in the game Fred and John were discussing.  

I don't have a sister. And isn't there another square that you can place the same piece and have the same result?

Edit: just realised why Bg3 doesn't work.

Bishop on h4.

Black has to move the bishop to avoid stalemate, and the it's a rookpawn and wrong coloured bishop ending.

The person who spoke of trolls in another thread.