Yes, for skills, it’s an automatic success, even on a Nat-1
You succeed on a skill check every time your roll plus your bonus meets or exceeds the DC. If your bonus is as large or larger than the DC, then there is no possible roll that results in you failing to meet the DC, and thus you automatically succeed.
The auto-failure on a natural-1 rule does not apply to skill checks (nor does the auto-success on a nat-20 rule).
As an important aside, there are a lot of very good reasons for natural-1 not applying to skill checks, and it is strongly recommended that you not houserule that particular feature.
Skills are supposed to be things you are reasonably good at, particularly when you’re trained and high level. A 5% chance of failure is much too high a rate for the kinds of activities we’re usually talking about. Worse, in some cases, like jumping, there is no particular set DC: you just go as far as you can based on what you roll. What does an “automatic success” even mean in that case? Does that mean I have a 5% chance of successfully jumping to the moon?
Even in less silly cases, training is supposed to represent a deep internalization of the skill. At appropriate levels of training, you are intended to be able to accomplish certain tasks – indicated by having a DC low enough that they can be hit automatically – as easily as the average person can sit down in a chair. People do, on rare occasions, mess up sitting in a chair, particularly in poor conditions (poorly-balanced chair, person is inebriated, whatever), but it happens way less frequently than 5% of the time, even in poor conditions. The d20 simply doesn’t have the granularity to measure that. Changing this is a severe nerf to the skill-based folks in the party, and causes some pretty serious problems.