Should be 300 Ohm
I guess the Max Ic = 1A in the data sheet of the transistor probably says that it should get a max of 1Amp on the Base Input?
No no no. Ic is the collector current, that's the current flowing from collector to emitter AND that is the maximum you can let it be before damaging the device. All transistors have a value of gain. That is the multiplying factor to calculate the current flowing in the base to that flowing in the collector. So if your transistor has a gain of 100 (typical value) you only need 1mA flowing in the base to get 100mA flowing from collector to emitter.
Think of the collector emitter junction of the transistor as being like a variable resistor, the thing that controls it is the base current. When a transistor is conducting as much as the collector load will allow it is said to be saturated (or the pot turned down to zero), more base current won't give you any more collector current.
With digital electronics like this we want the transistor to be saturated, we say it is turned on. So you see only a few mA are needed to turn a transistor on and hence a resistor is put in the base to limit the current.
Too much base current can destroy the transistor, the data sheet will say how much this is. If the transistor is too saturated (with base current) then when the base current is removed (transistor is switched off) there will be a delay in switching off and the collector current will slowly drop (not rapidly like you want), this is called the charge storage effect.
In fact you often see 1K in the base, with 5V this gives you a base current of 5mA, which given a gain of 100 will only not saturate at collector currents of 500mA. Which is good enough most of the time.
Some power transistors (lots of Ic capability) have low gains some down to about 10, then you might need a second transistor to drive the first.
Hope that makes sense. 8-)