SAN JOSE MINE, Chile — A powerful rescue drill broke through to the deep underground chamber where 33 miners have been trapped for more than two months, signaling that their deliverance is closer than ever at hand.

The government set Wednesday as its target date to begin hoisting the trapped men out of the mine, saying that the nearly half-mile long shaft is sufficiently sound that only a small stretch of it will need to be reinforced with steel pipe.

The...