I would recommend Walter Rudin's classic text entitled Real and Complex Analysis for the mathematically mature undergraduate student. (Of course, this should really only be read after one has familiarity with most of Rudin's earlier textbook: Principles of Mathematical Analysis.)
The beautiful thing about this book is that nearly every example or result stated by Rudin is used in "the big picture". As you progress through the book, you really start to appreciate the magic Rudin has used to weave together analysis in a manner that is rarely done in other textbooks.
While Rudin states in his preface that the textbook is intended as a first year graduate course on the subject of analysis, I believe that it is quite possible for a mathematically mature undergraduate to follow it. There is no assumption in the book that the reader has any familiarity with linear algebra, abstract algebra, general topology etc. beyond that which was covered in the first seven chapters of his earlier book, but realistically one would like to have at least mastered the basics of these areas before attempting to delve deeper into the analysis covered in this book. (There is a chapter on Banach algebras, but all the necessary abstract algebra here is developed from scratch.)
The textbook is indeed challenging with plenty of exercises, but it is not something with which a student with the correct prerequisites should have tremendous difficulty. Furthermore, if a student can successfully read most of the book, he is well-equipped to go deeper into most branches of modern analysis, and should find the other classic texts in the subject, for instance Royden's Real Analysis, Bartle's The Elements of Integration and Lebesgue Measure etc., very easy to follow.
The Amazon page for this book can be found here.