Short answer: No
Only these cards are supported:
Catalyst 6.5: ATI Radeon 7000, 7200, 7500, 8500, 9000, 9100,
9200, 9250, 9500, 9550, 9600, 9700,
9800, X300, X550, X600, X700, X800,
X850, X1300, X1600, X1800, X1900,
X1950, ArcadeVGA 9200/9250, etc.
Catalyst 9.3: ATI Radeon 9500, 9550, 9600, 9700, 9800, X300,
X550, X600, X700, X740, X800, X850,
X1050, X1200, X1300, X1550, X1600,
X1650, X1800, X1900, X1950, HD 2350,
HD 2400, HD 2600, HD 2900, HD 3200,
HD 3300, HD 3400, HD 3410, HD 3450,
HD 3550, HD 3570, HD 3600, HD 3610,
HD 3690, HD 3730, HD 3750, HD 3800,
HD 3830, HD 3850, HD 3870, HD 4230,
HD 4250, HD 4350, HD 4550, HD 4570,
HD 4580, HD 4650, HD 4670, HD 4730,
HD 4750, HD 4800, HD 4850, HD 4870,
HD 4890, etc.
However, you can use your HD 5xxx card just fine under Linux (GroovyArcade) in case you want to try.