That is correct. Glaciers melt constantly, new snow forms as moisture accumulates in higher elevations and condenses, falling as snow. The snow becomes the beginning of the glacier. It's a cycle that repeats. It is repeating right now, every year.

https://nsidc.org/cryosphere/glaciers/life-glacier.html

https://www.iceagenow.com/List_of_Expanding_Glaciers.htm

http://www.esa.int/spaceinimages/Images/2016/11/Pio_XI_Glaci...