We have developed a technique for producing ultracold beams of ground state [sup 4]He atoms by evaporation from a film at subkelvin temperature in a dilution refrigerator. Our analysis shows that we have cooled atoms to below 700 [mu]K and that the cooling process can be continued to below 100 [mu]K. This new technique has great promise for determining the He-He interaction potential to high precision and searching for [sup 4]He dimer formation. It might also be used to study quantum reflection off surfaces and to search for Bose-Einstein condensation in a dilute gas of [sup 4]He.