"We will show evidence that will solve her disappearance," Gillespie said in a telephone interview.
The National Transportation Safety Board has been examining some of the personal effects and documents, but the panel has not yet verified any of the evidence.
Earhart was the first woman to fly across the Atlantic and set a number of aviation speed and altitude records, making her a hero and celebrity of the 1930s.
Accompanied by navigator Fred Noonan, she was attempting to become the first woman to fly across the Pacific when her plane disappeared on July 2, 1937. Neither was ever found.
Gillespie theorizes that Earhart, looking for Howland Island, found herself low on fuel near the island of Nikumaroro and landed there.
He told the Houston Post that his group found part of the fuselage of Earhart's twin-engine Lockheed 10-E Electra and personal effects in the jungle on Nikumaroro last fall.
The island is a 3 1/2-mile-long-by-half-mile-wide atoll of tropical vegetation and coconut palm trees in the island nation of Kiribati.
One of the effects was reported to be the remains of an American-made size 9 shoe, the size worn by Earhart. Gillespie said a shoe expert has been examining the evidence.
"It has been a long, difficult, expensive project," Gillespie told the Houston Post.
The four-year project, he said, cost $750,000, all raised through public contributions.
The Post did not explain how the plane was found. But last June, Oceaneering International, a Houston-based company, told Reuters news service it had been hired by the nonprofit TIGHAR to use a highly sophisticated underwater sonar device to scan the ocean bottom for remains.
The company employed the same technology to find pieces of the space shuttle Challenger in the Atlantic Ocean after it exploded shortly after takeoff in January 1986. The technology was used again in 1990 to find a door that fell from a United Air Lines jet into water 14,400 feet deep in the Pacific.
Gillespie told Reuters last summer that TIGHAR's theory was that Earhart and Noonan got lost after they took off from Lae, New Guinea, headed for Howland Island and ran out of fuel.
He said that Earhart probably made an emergency landing on the island, and that she and Noonan may have survived the landing for a short period thereafter. In 1938, a severe drought struck the area.
A 1989 TIGHAR expedition to Nikumaroro found an aluminum box with navigational reference materials. The box appeared to be similar to one in an old photograph of the ill-fated plane, Gillespie said.
When Earhart's plane disappeared, President Franklin Delano Roosevelt ordered a search by more than 100 planes and six ships. At least 77 radio transmissions from Earhart were reported over three days.
Records of those signals were used by navigators in 1986 to plot the location of the wreck.