One can find a considerable amount of material on the Internet. However, none of it gives a definite answer to your query. On the contrary, there is much speculation regarding the final resting place of the ark and whether it still exists.
For those interested in learning more about the Ark of the Covenant, I'd like to give a brief history. As described in various Old Testament texts, commissioned by God, it was made of acacia wood with gold plating, about 45 inches wide and 27 inches high and long.
The upper surface or mercy seat of God had a rim of gold with two cherubim facing each other with their gold wings covering the box. It was "the glory of Israel" (Lamentations 2:1)
Jewish faith
It was built to house the stone tablets of the Ten Commandments which were signs of God's covenant (Deuteronomy 31:26).
The ark was the sign of God's presence and saving power. As the embodiment of God, it was the cornerstone of the evolving Jewish faith.
The ark was carried before the people in their journeys (Numbers 4:5-6 and 10:33-36). In Joshua, we read that it opened a path through the Jordan for the people to pass through (3:15-16) and was carried around Jericho when the walls came down (6:4, 6, 8).
Taken into a battle to assure victory, it was captured by the Philistines (1 Samuel 4:3-11) who later returned it because it brought them bad luck (1 Samuel 5:7-8).
In a grand procession, David had the ark brought to Jerusalem. Solomon built the first Temple and placed the ark there (1 Kings 8:6-9).
Based on historical documentation, a well-known archaeologist believes he has identified the original location of the ark in the Holy of Holies under the Muslim Dome of the Rock.
When the Babylonians later destroyed and plundered the Temple, the ark disappeared with no trace.
Various theories have been proposed about the present location of the ark, if it still exists.
Where is it?
One posits that it was hidden in a secret room under the Dome of the Rock.
However, no one is allowed to investigate this possibility because of political consequences. Another suggests that it is hidden in a cave near the Dead Sea.
No evidence of the ark has been found there.
A popular theory, as suggested by your question, is that it was secreted out of the Temple to Axum, Ethiopia.
The story goes that when the Queen of Sheba, who according to their tradition, was an Ethiopian queen, visited Solomon in Jerusalem, she returned home carrying within her Solomon's son.
Twenty years later, Melenik went to see his father but because of jealousy in the court, he was sent back to his mother, accompanied by the first-born sons of the elders.
One of them took the ark and when Melenik found out, he felt that it must have been God's will and so the ark settled in Axum, Ethiopia.
The ark is kept in Saint Mary of Zion Church built by Emperor Haile Sellassie.
The original church, constructed in the fourth century when the Axumites converted to Christianity, was destroyed by Muslims in 1535.
However, the ark had been taken to a secure hiding place and remained there until the new church was built in 1965.
Briton Graham Hancock, claims he visited Axum and spoke to the old monk who guards the ark and who testified to seeing it.
However, Hancock was not allowed to see it.
Is this story legend or history? No one really knows. Since no one can see it, obviously no scientific methods can be used to examine it.
What is intriguing is that, until the time of Solomon (970-930 BC), the ark is mentioned in the Bible more than 200 times but it is not mentioned again. How can it be possible that such a valuable treasure for the ancient Israelites can just disappear without lamentation and mourning, without searching and re-building?
But that should not surprise us, for Jeremiah says, "When you multiply and become fruitful in the land, you will no longer say 'The Ark of the Covenant of the Lord!'
"You will no longer think of it or remember it or miss it or make another.
"At that time, you will call Jerusalem the Lord's throne.
"There, all nations shall be gathered together to honour the name of the Lord" (Jeremiah 3:16-17).
Therefore, the elementary cult of the ark by nomadic tribes was meant to be replaced by a more spiritual and universal presence of God in Jerusalem.
Christian meaning?
Does the ark have meaning for Christians?
The Book of Revelation tells us "God's temple in heaven opened and there could be seen the Ark of God's Covenant.
"A great sign appeared in the sky, a woman clothed with the sun. . . . She was with child. . . . She gave birth to a son - a boy destined to shepherd all nations" (Revelation 11:19; 12:1, 2, 5).
God's covenant with the Israelites, symbolized by the ark, has been fulfilled in the coming of Christ.
God's presence on earth in the person of Jesus and the Holy Spirit has brought God's saving power, new life and light to all.
Why do Christians and Jews not demand the examination of the oldest relic on earth, namely, the Ark of the Covenant in Axum, Ethiopia, as was done for the Shroud of Turin? Please shed some light on this, if you can.