For the motor, a stall current of 1A means that at stall, supplying 6V will cause the motor to draw 1A. The no-load current of 250mA means that supplying 6V will cause the motor to draw 250mA.
The half-bridge IC you have given has no current overload protection. Its peak current of 1.2A is high enough, but the 600mA continuous current is not. This can normally cause it to overheat, and possibly break. Under the absolute maximum ratings, it has a peak current of 1.2A for <100us. Normally, it may be possible to heat an IC a little more than recommended if a good heatsink is used. However, the <100us implies that this is not possible, because the heat would not be able to transmit through the package fast enough.
These are your options:
You can supply your own current foldback protection. By using a current sensor of some sort, you can detect when there is too much current flow, you can then scale down the voltage, possibly down to $6\times0.6\div1 = 3.6V$. Assuming a fast control circuit, this is probably fine (due to the peak current allowed).
You can run the motor at only 3.6V. This limits the stall current, torque and maximum no-load speed, but means that you do not have to do any current foldback.
If you can measure the speed of the motor, and assuming you know the back EMF constant of your motor, you can convert speed to back EMF voltage. This is the additional voltage that you can supply to your motor (in addition to the 3.6V which gives 0.6A of current). This is because the amount of the back EMF voltage represents the voltage which will cause zero current to flow.
Note that when you run the motor at 3.6V, this is just the average voltage. Assuming you use pulse width modulation, this is equivalent to a power supply of 6V, with a duty cycle of 60%. Of course, the actual supply voltage must be within the operating condition of the device (36V).
You might be tempted to use two sets of half-bridges for each direction of the same motor. However, if you do this, you should be aware that it is quite difficult to control how much of the current comes from each parallel supply. Therefore, due to slight differences in transistor characteristics, one will supply more current than the other.
The specifications do not appear to mention anything about compensating current with two motors.
Why don't you just use the L293 quad half-bridge drivers (which allow 1A continuous)?