Is there any hope for writing good code atop a horribly designed database?

Question

Here's my predicament. One of several programs I've recently inherited is built with a horrible database on the backend. The esteemed creators of it apparently did not appreciate relational concepts. A table for each and every client, named as a unique client ID. Eighty-three cryptically named fields. The code is all procedural with dozens of concatenated inline SQL statements.

As we weren't provided with an important ancillary application that runs off the same database, I've been tasked with recreating it from scratch. I'm a sole developer, which isn't even my primary responsibility as at least half of my time is taken up by operations stuff. There's an unavoidable deadline set for 30 days from now.

Despite my inexperience, I'm certain I could have designed this database and existing application much better than they were, but I don't really think it's realistic for me to alter the database, adjust the existing application, and be sure I didn't break anything while needing to create the additional application this quickly.

So let's assume I am stuck with the terrible database. Needing to work with such a bad structure, would anything I write that conforms to it just add to the heaping pile of technical debt to be shelved away until something completely breaks or new functionality is needed? How could I approach this situation and get something good out of it besides a hopefully functional application?

edit: In case anyone's interested, we ended up scrapping this horrible database and the application that ran on it. We outsourced the creation of the ancillary application (I wasn't involved in setting this up) to ultimately two different contractors who both ended up falling through on us, accomplishing nothing. I ended up having to rush out a horrific, partially functional hack of a fix in three days that's still in use today.