An alternative given by [1] is to compute the interval for the logit AUC:
$ log \left( \frac{AUC}{1-AUC} \right) \pm \phi ^{-1} \left( 1 - \frac{\alpha}{2} \right) \frac{\sqrt{AUC}}{AUC(1 - AUC)} $
so that you get an asymmetric interval. In your case, you would get a 95% CI $(0.38, 0.81)$.
If you are frequently dealing with high AUCs and small sample sizes, you may want to have a look at [2] that shows there is no single method that can optimally compute confidence interval for all ROC curves.
[1] Pepe MS, The Statistical Evaluation of Medical Tests for Classification and Prediction, OUP 2003, p. 107
[2] Obuchowski NA, Lieber ML, Confidence bounds when the estimated ROC area is 1.0, Acad Radiol. 2002, 9 (5) p. 526-30
The question is quite vague so I am going to assume you want to choose an appropriate performance measure to compare different models. For a good overview of the key differences between ROC and PR curves, you can refer to the following paper: The Relationship Between Precision-Recall and ROC Curves by Davis and Goadrich.
To quote Davis and Goadrich:
However, when dealing with highly skewed datasets, Precision-Recall (PR) curves give a more informative picture of an algorithm's performance.
ROC curves plot FPR vs TPR. To be more explicit:
$$FPR = \frac{FP}{FP+TN}, \quad TPR=\frac{TP}{TP+FN}.$$
PR curves plot precision versus recall (FPR), or more explicitly:
$$recall = \frac{TP}{TP+FN} = TPR,\quad precision = \frac{TP}{TP+FP}$$
Precision is directly influenced by class (im)balance since $FP$ is affected, whereas TPR only depends on positives. This is why ROC curves do not capture such effects.
Precision-recall curves are better to highlight differences between models for highly imbalanced data sets. If you want to compare different models in imbalanced settings, area under the PR curve will likely exhibit larger differences than area under the ROC curve.
That said, ROC curves are much more common (even if they are less suited). Depending on your audience, ROC curves may be the lingua franca so using those is probably the safer choice. If one model completely dominates another in PR space (e.g. always have higher precision over the entire recall range), it will also dominate in ROC space. If the curves cross in either space they will also cross in the other. In other words, the main conclusions will be similar no matter which curve you use.
Shameless advertisement. As an additional example, you could have a look at one of my papers in which I report both ROC and PR curves in an imbalanced setting. Figure 3 contains ROC and PR curves for identical models, clearly showing the difference between the two. To compare area under the PR versus area under ROC you can compare tables 1-2 (AUPR) and tables 3-4 (AUROC) where you can see that AUPR shows much larger differences between individual models than AUROC. This emphasizes the suitability of PR curves once more.
Best Answer
In your situation it would be fine to plot a ROC curve, and to calculate the area under that curve, but this should be thought of as supplemental to your main analysis, rather than the primary analysis itself. Instead, you want to fit a logistic regression model.
The logistic regression model will come standard with a test of the model as a whole. (Actually, since you have only one variable, that p-value will be the same as the p-value for your test result variable.) That p-value is the one you are after. The model will allow you to calculate the predicted probability of an observation being diseased. A Receiver Operating Characteristic tells you how the sensitivity and specificity will trade off, if you use different thresholds to convert the predicted probability into a predicted classification. Since the predicted probability will be a function of your test result variable, it is also telling you how they trade off if you use different test result values as your threshold.
If you are not terribly familiar with logistic regression there are some resources available on the internet (besides the Wikipedia page linked above):
R, the UCLA stats help website is generally excellent and has a relevant page here.