Yes, It is absolutely safe and smooth, either by bus/car which is going to take between 3-4 hours or by airplane. Usually, people prefers to rent cars with a driver and this takes 4 passengers and the ticket is cheap around 10 Euros (per passenger) and it is faster way of getting to the required destination Amman-Damascus or vice versa. about 8 years ago
Jackie, getting a driver is a good idea. The roads can be in poor condition, and the driving just as bad. Jordan has a pretty high road death toll. It's recommended you don't travel at night, throw in wandering livestock, broken down vehicles and poor lighting into the mix. A local driver will be better practised at these hazards. Plus, a driver is always guilty if they hit a pedestrian - by law - so you don't want any possibility of being involved in that unpleasantness.
Just on another travel safety related issue.... keep an eye on the political situation in Jordan and especially Syria. The "egypt effect" has reached those countries. There've been protests for 4 days in Darar in the south of Syria, and increasing restlessness in Damascus. The Syrian government cracks down pretty hard on protesters. 4 people have died and hundreds injured in recent protests.
If it's a working trip, make sure your employer has security precautions in place and that they have appropriate insurance for you (travel insurance doesn't cover people in paid work).
Should be a great trip, but you need to be on your guard a bit. good luck. Phil about 8 years ago