application note adc0804
I see your problem with a wire board instead of a PCB.
Your temperature sensor is an analog device. Its output is converted to digital by your ADC chip. Any components through which your analog signal passes should be connected to AGND. Any components throught which your digital signals pass should be connected to DGND. Both of those grounds should be connected together at the supply point.
On a PCB, you would connect the two pins to the same plane. However the section of the plane for the analog grounds (AGND) would be segregated from the DGND as shown in the articles I referenced above.
The goal is to have all analog current flow only through analog sections of the ground, and all digital current flow only through digital sections of the ground. The two should not share ground return paths.
Ground loop refers to the ground side of the complete signal path. Keep in mind that all signals flow through whatever conductor you have provided for them - AND THEN have to return via the ground connections. Every signal is a loop - from source to sink, and back to source (via the ground connection). The loops should be kept as small as possible to prevent inductive coupling of noise. The larger the loop, the larger the inductance, and the more coupling you get for undesirable signals. Short, direct, connections are the way to do it with a wired board. You can run a ground wire from the supply to a central point, and connect your grounds to that point - that's called a "star" ground scheme.
In your case, you would want two "stars". One for DGND and one for AGND. Then the two would be joined at the power source.