By Steven Scheer and Lefteris PapadimasJERUSALEM/ATHENS, Aug 31 (Reuters) - Israel and Greece signed a €3 billion ($3.5 billion) defence deal on Monday under which Israel will build a multi-layered air defence system for Greece as Athens pushes ahead with a broader military modernisation programme.Israel's defence ministry said it was Israel's largest-ever defence export deal with Greece and one of the biggest in the country's history. The agreement confirmed an earlier Reuters report. Amir Baram, director general of Israel's defence ministry, said the so-called Achilles Shield deal reflected "Israel's technological edge and the proven track record of our defence systems in the field."Under the deal, Israel will supply Greece with the David's Sling and SPYDER systems produced by Rafael Advanced Defense Systems and the BARAK MX system made by Israel Aerospace Industries, as well as MMR multi-mission radars for air surveillance, manufactured by IAI unit ELTA Systems.David's Sling was used by Israel to intercept Iranian missiles during the war with Iran this year. Greece becomes the second customer for the system after Finland.Greece spends nearly 3.5% of its gross domestic product on defence, a higher proportion than many NATO allies due to its long-standing dispute with neighbouring Turkey.Athens plans to spend about €28 billion by 2036 to buildthe Achilles Shield multi-layer anti-ballistic, anti-aircraft and anti-drone system, and to buy up to 40 F-35 fighter jets from the U.S. and frigates from France and Italy.Under Monday's deal, Greece will also receive a new national command-and-control system, developed by Rafael, to integrate the country's air-defence capabilities. A supplementary €26 million agreement will see Israel supply Greece with Rafael's Drone Dome systems to protect strategic sites against drones and reinforce existing defences.Israel will also expand defence-industrial cooperation with Greece, including transfers of technology and expertise to local industry.($1 = 0.8623 euros) (Reporting by Steven Scheer. Editing by Alison Williams and Mark Potter)