Mark of Gideon
The planet Gideon appears to be a haven — the inhabitants are healthy and no one seems to ever die in the totally germ-free environment.
The United Federation of Planets sends the U.S.S. Enterprise to Gideon, in the hope that the Gideons will accede and become a member. Reluctantly, the Gideon council allows Captain Kirk alone to beam to their council chamber, and he transports off the ship. When Kirk apparently fails to arrive on the planet, the Gideon council refuses to allow more people on the surface, even for a search party.
Kirk finds himself on his own ship, where all of his crew have seemingly vanished. After searching the ship, he finds one lone, beautiful woman, Odona. Kirk determines that Odona is from Gideon, and that they are on a false ship, built on the planet's surface. When Odona becomes ill, the Gideon's plan is revealed: using Kirk's blood, Odona was infected with a disease which Kirk had recovered from — Vegan choriomeningitis. The infection is fatal, and the Gideons hoped to spread it across their world to reduce the population.
Spock beams down, discovers the false Enterprise, and returns to the real starship with Kirk and Odona, who — while cured — is still able to infect her people. Odona, the daughter of Gideon Council leader Hodan, is returned to her people delighted that she will cause a lethal plague to reduce the overpopulation.