Synthetical formation of diamonds
Natural Diamonds are formed under high pressure and temperature at depths of 140 to 190 kilometers (87 to 118 mi) in the Earth's mantle. Carbon-containing minerals provide the carbon source, and the growth occurs over periods from 1 billion to 3.3 billion years (25% to 75% of the age of the Earth). Diamonds are brought close to the Earth's surface through deep volcanic eruptions by a magma, which cools into igneous rocks known as kimberlites and lamproites.

According to Wikipedia,


  The formation of natural diamond requires very specific conditions—exposure of carbon-bearing materials to high pressure, ranging approximately between 45 and 60 kilobars (4.5 and 6 GPa), but at a comparatively low temperature range between approximately 900 and 1,300 °C (1,650 and 2,370 °F). These conditions are met in two places on Earth; in the lithospheric mantle below relatively stable continental plates, and at the site of a meteorite strike.


So yes, Diamonds can be produced synthetically if you can simulate almost similar conditions as in Earth's Mantle. (See high-pressure high-temperature process; Synthetic Diamond) An alternative, and completely different growth technique is chemical vapor deposition (CVD).