Verizon Wireless said Friday it will allow its customers to opt out of an advertiser-friendly method of tracking their Web visits. The telecommunications provider has been employing a unique identifier for each customer since 2012, a so-called supercookie it uses to target advertising.

Verizon came under fire from privacy advocates late last year for the practice, which they said could expose customers’ Internet histories to both government spies and hackers. Verizon did not say when it would begin allowing customers to kill the supercookies, but the New York Times cited a company statement by representative Debi Lewis saying the change came as a result of listening to customer feedback:

Verizon takes customer privacy seriously and it is a central consideration as we develop new products and services. As the mobile advertising ecosystem evolves, and our advertising business grows, delivering solutions with best-in-class privacy protections remains our focus. We listen to our customers and provide them the ability to opt out of our advertising programs. We have begun working to expand the opt-out to include the identifier referred to as the UIDH, and expect that to be available soon. As a reminder, Verizon never shares customer information with third parties as part of our advertising programs.

The supercookie is actually a unique identifier header that can’t be evaded by using “incognito” or “private” mode on Web browsers. The supercookie collects data on Web traffic for an individual customer and builds a pattern of interests and activities that would then be used for targeted advertising: Ad agencies pay more for these ads than they do for standard ads.

Verizon had allowed customers to remove themselves from the targeted advertising, but continued to track them with the supercookies. In November, it defended supercookie tracking by saying it notified all customers of the practice. However, the Electronic Frontier Foundation, a privacy-advocacy group that focuses on the Internet, said Verizon kept its users in the dark about the program by burying their notifications in long-form user agreements.

Verizon’s decision came hours after it received a letter from the U.S. Senate’s Committee on Commerce, Science and Transportation that criticized the practice and demanded an explanation for it. The senators said Verizon was “doing nothing to stop third parties from exploiting” the supercookies.