From Wikipedia:
Before the Avignon Papacy (1305-1378), the headquarters of the Holy See were located at the Lateran Palace.
Every pope from Miltiades occupied the Lateran Palace until the reign of the French Pope Clement V, who in 1309 decided to transfer the official seat of the Catholic Church to Avignon, a papal fief that was an enclave within France. During the Avignon papacy, the Lateran Palace and the basilica began to decline. Two destructive fires rampaged through the Lateran Palace and the basilica, in 1307 and again in 1361. In both cases, the Avignon papacy sent money to their bishops in Rome to cover the costs of reconstruction and maintenance. Despite the action, the Lateran Palace and the basilica lost their former splendor.
Although, presumably the Pope would have left artifacts at the Lateran Palace, probably in safe storage in the papal tombs, underneath - how historically acurate do you want it to be? If it is a real artifact I can pin down it's position, if it's something you've made up for the story, it can be where-ever you want it to be. The logical place would be in the tombs, safe from any fires, or in Avignon.
When the Avignon papacy formally ended and the Bishop of Rome again resided in Rome, the Lateran Palace and the basilica were deemed inadequate considering the accumulated damage. The popes took up residency at the Basilica di Santa Maria in Trastevere and later at the Basilica di Santa Maria Maggiore. Eventually, the Palace of the Vatican was constructed, and the papacy moved in; the papacy remains there today....
After the Avignon Papacy the church administration moved to Vatican Hill and the papal palace was (until 1871) the Quirinal Palace, upon the Quirinal Hill....The palace was built in 1573 by Pope Gregory XIII, - so too late for your piece, Sam. Also the Vatican Museum was not set up until the 16th Century.
- NaomiM
<Added>http://en.wikipedia.org/wiki/Basilica_di_San_Giovanni_in_Laterano<Added>This second church lasted for four hundred years and then burnt in 1308. It was rebuilt by Pope Clement V and Pope John XXII, only to be burnt down once more in 1360, but again rebuilt by Pope Urban V.
Since your date is 1315, soon after the fire of 1308, reconstruction would probably not have been completed and your 'thief' could enter in the guise of eg. a builder or stonemason.
<Added>Unfortunately, I can't find reference to papal tombs in a catacombs-type set up. Today they are simply stone tombs inside the current building:
Papal Tombs
There are six papal tombs inside the basilica: Alexander III (right aisles), Sergius IV (right aisles), Clement XIII Corsini (left aisle), Martin V (in front of the confessio) by Simone Ghini; Innocent III (right transept); and Leo XIII (left transept), by G. Tadolini (1907).
<Added>Probably better to base it in Avignon. Unfortunately the Gothic palace was not built until 1335:
http://en.wikipedia.org/wiki/Palais_des_Papes
The Palais des Papes in Avignon, France is one of the largest and most important medieval Gothic buildings in Europe. It is one of many places called the Palace of the Popes.
Avignon became the residence of the Popes in 1309, fleeing the violent chaos of Rome. The Palais was built between 1335 and 1364 on a natural rocky outcrop at the northern edge of Avignon, overlooking the river Rhône. The site was formerly occupied by the old episcopal palace of the bishops of Avignon.
<Added>
Perhaps you should look at Cathédrale Notre-Dame des Doms, Avignon
Cathédrale Notre-Dame des Doms (Avignon, Provence): Next to the Palais des Papes, this was a luminous Romanesque structure before baroque artists took over. It was partially reconstructed from the 14th through the 17th century. In 1859, it was topped by a tall gilded statue of the Virgin, which earned it harsh criticism from many architectural critics. The cathedral houses the tombs of two popes, John XXII and Benedict XII. You'd think this cathedral would be more impressive because of its role in papal history, but it appears that far more time and money went into the construction of the papal palace. Nevertheless, the cathedral reigned during the heyday of Avignon.
<Added>Of course, with all this later rebuilding, who is to say there weren't catacombs below these buildings at one time. But probably of Roman design rather than a Gothic crypt which came later in the 14th Century.
Although from Wikipedia:
http://en.wikipedia.org/wiki/Catacombs_of_Rome
By the 10th century catacombs were practically abandoned, and holy relics were transferred to above-ground basilicas. In the intervening centuries they remained forgotten until they were accidentally rediscovered in 1578, after which Antonio Bosio spent decades exploring and researching them for his volume, Roma Sotterranea (1632).
<Added>The accidental discovery in 1578 of an ancient subterranean cemetery on the Via Salaria had attracted general attention in Rome
http://chi.gospelcom.net/DAILYF/2001/05/daily-05-31-2001.shtml<Added>So, by 1315 the last pope would have died 1314, the new palace in Avignon would not have been built, the old one in Rome would be nearing completion after the fire in 1308. A relic would probably have been stored in one of the papal basilicas in Rome or the Cathédrale Notre-Dame des Doms, Avignon, unless it dated back to before the 10th Century, in which case it might still be down the old roman catacombs under the city.
<Added>One last thought:
If the artifact is anything to do with a saint - relic or bone - it is a little different since there was a huge pilgrimage industry based around them. They were kept on display to entice the pilgrims to visit - and many churches and cathedrals petitioned the Vatican for pieces of saints so as to attact their own pilgrims...