By Hugh Bronstein

BOGOTA, July 12 (Reuters) - A Colombian businessman kidnapped by leftist rebels died in a military rescue attempt on Thursday, illustrating the dilemma the government faces over thousands of hostages languishing in secret guerrilla jails.

The death of Diego Mejia hit a nerve in Colombia, where French-Colombian politician Ingrid Betancourt and three Americans are among captives held by the Revolutionary Armed Forces of Colombia, or FARC.

"Mr. Mejia was injured when we tried to take the area and died when we tried to evacuate him," army chief Mario Montoya said.

President Alvaro Uribe is being implored by the families of many kidnap victims not to attempt dangerous rescues and to grant the FARC's conditions for starting hostage exchange negotiations.

But Uribe has signaled he will stick to the hard-line security policies that got him re-elected last year, refusing a FARC demand that he demilitarize a western rural area to be used for talks aimed at swapping rebel hostages for guerrillas held in government jails.

The FARC kidnaps for ransom and political leverage in its 4-decade-old war against the state. They captured Mejia in May and were holding him in the central coffee-growing province of Caldas.

Betancourt was taken during her 2002 presidential campaign. U.S. defense contractors Thomas Howes, Keith Stansell and Marc Gonsalves were seized during a 2003 anti-drug mission.

Hundreds of thousands of Colombians took to the streets last week protesting the June deaths of 11 provincial lawmakers captured by the guerrillas in 2002.

The FARC says the lawmakers died in crossfire when an unidentified military group attacked the jungle prison where they were being held. Uribe accuses the guerrillas of murdering the men, saying no soldiers were in the area at the time.

Colombia was stunned in 2002 when the guerrillas kidnapped the lawmakers from a government building in the western city of Cali by masquerading as soldiers and escorting them onto a bus, saying there was a bomb in the building.

The country is considered the world's kidnapping capital, with about 25,000 cases registered over the last decade.

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