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Joseph, born "Yousef", was the third of six children, born to Massoud Yousef Costa and Saada Simaan Ghantous. Costa's father was a gambler who experienced frequent finanial difficulties, and as a teenager Costa worked as a tailor, creating custom-made clothing to help his family financially.
In 1956, he believed he was called to the ministry. With the help of Dr. James Ragland, a Southern Baptist missionary working in Lebanon at the time, he planted a church in the Ras Beirut area of the city of Beirut. In 1959, he was ordained as the pastor of the new church. Having finished his sophomore year at Haigazian College, he then went to the Arab Baptist Theological Seminary to pursue a diploma in theology.
He was elected president of the Lebanese Baptist Convention in 1963-1965. In 1970, on the thirteenth anniversary of his selection as pastor of the Ras Beirut Baptist Church, he resigned, returning to Haigazian College to complete a B.A. in English literature, which he completed in 1971. He oversaw the work of Clarion Publishing House until December 1975, when he left to escape the Lebanese civil war. During the first few months of the war, the family home was looted and bombed. He took his family to Jordan, from where the whole family migrated to the United States in 1976. After living in Mobile, Alabama, for a few months after arrival, he moved to the Northeast and pastored an evangelical group of Arabic-speaking Christians in the Philadelphia metropolitan area. He lived in south New Jersey until his death.
During his life he translated a number of books into Arabic, including the musical Christmas cantata Night of Miracles by John W. Peterson. He translated many of the classical hymns into Arabic such as, "I Don't Know About Tomorrow", "On a Hill Far Away" and "Cum Bya My Lord". He also wrote a number of books, including Apples of Gold (two volumes), Explosive Letters, Personalities of the Bible, Removing the Mask, Christ is Coming Again, Marriage: Heaven or Hell, and The Bible: Is It The Word of God?.
He was a regular conference speaker in the US, Canada, Australia and the Middle East. He was heard weekly on The Hour of the Harvest, the Arabic broadcast of LifeWord, a ministry of the Baptist Missionary Association of America. The broadcast was heard in the Middle East on Trans World Radio, broadcasting from Monaco and Cyprus from 1977 to 1989, and currently[when?] on radio station HCJB.
Costa died on May 8, 1989 from complications related to cancer and is buried in Blackwood, New Jersey.
On June 7, 1957, he married Nawal Anis Haddad. They had four sons, Charles (1958), William (1961), Dori (1967) and Roy (1972).