My mother has AT & T stock certificate noting 10 shares and dated August 12, 1968. How can I verify that this certificate is of value. She is 90 years old & doesn't remember if these certificates were exchanged.

Wow - AT&T has certainly been through many, many changes since 1967!

As AT&T Corp (the old AT&T, or "Ma Bell") spun off the seven Baby Bells in 1984, then Lucent and NCR in 1996, had a 3:2 stock split in 1999, spun off AT&T Wireless in 2001, spun off AT&T Broadband and did a 1:5 reverse stock split in 2002, before finally being bought by SBC, which then changed its name to "AT&T Inc" - the result is your grandmother's original 10 shares in AT&T Corp from 43 years ago I believe is now equivalent to 2.33826 shares of the new AT&T Inc. The value of that fractional share should have been paid to her in cash in 2005, leaving her with 2 shares of AT&T Inc. Her cost basis is 24.28% of her original purchase basis (the rest of the cost basis was transferred into her stock holdings in all the various spin-offs).

Heres' a handy calculator to help you see how AT&T's history affects her stock basis and number of shares: Cost Basis Worksheets for AT&T Corp.

She (or you) should contact AT&T Inc investor services to ask about converting those old AT&T Corp shares into new AT&T Inc shares. Contact information is here:
Stockholder Inquiries - Investor Relations
Good luck!