But how can just one signal determine the ship's location? Doesn't that just say how far the ship is from Earth?
tl;dr: The atomic clock makes one part of the measurement a little easier, but navigation is still done with some combination of range-rate, Delta-DOR and calculation of gravitational effects using numerical integration and solar system ephemerides.
Navigation of spacecraft in deep space is done with several tools. Three of the main ones are:
Range-Rate is the two-way measurement of signals broadcast from one location on Earth, received and retransmitted by the spacecraft in a frequency-coherent way and received back on Earth. It's also called delay-Doppler. You get a range from the delay, and a rate-of-change of range (a 1 dimensional velocity) from the Doppler shift.
Delta-DOR: is a one-way measurement of signals broadcast from a spacecraft and received by two widely separated deep-space ground stations on earth and the difference in the times of signal arrival is precisely measured (and used to calculate a bearing). This is corrected using information about the current delays due to Earth’s atmosphere, obtained by simultaneously tracking (from each ground location) radio signals from a quasar (within 10 degrees of the same direction).
Numerical Integration In spacecraft navigation you would use the above two methods plus extremely accurate simulations of the history of the spacecraft's trajectory based on all known forces, gravitational and otherwise, to build a complete picture of the trajectory of the spacecraft. Masses and positions of all large solar system objects already exist as ephemerides and these are inputs to such a calculation.
- Range-Rate gives a distance and speed without an accurate direction
- Delta-DOR gives an accurate direction
- Numerical integration augments these two and completes the picture, providing more accuracy than possible with the first two alone
How does an onboard atomic clock help interplanetary navigation?
@DavidHammen's excellent answer explains how having an atomic clock aboard a spacecraft allows the range-rate measurement to be done one-way instead of two way. I recommend you read the full answer, but briefly, the spacecraft broadcasts a signal encoded with time information from the atomic clock. These signals are received on Earth and compared to known time here, and the difference, once corrected for relativistic effects gives the distance. Doppler shift of the time signals or the frequency (if locked to the atomic clock) gives the rate or relative velocity.